unit 6
6. write a definite integral that is equivalent to lim n→∞ k=1En (3/n)(-2 + 3k/n)^4
S[-2,1] x^4dx
look at pages 1 and 2 for the graph problems on 6.2
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look at topic 6.1 for the graph problems
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look at #2 on topic 6.2 paper to find the table
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look at topic 6.4 for both pages
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look at the entire practice of 6.2 for the graphs
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look at #7 on the test prep of 6.1 to find the graph problem
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look at the top of page 1 of 6.2 to find the graph problem
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look at 6.6 test prep for the graph
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look at the first page of warm up to find the graph problem
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look at #1 of 6.6 practice to find the graph
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look at 6.5 on unit 6 review 6.1 to 6.5 for the back page with the graph
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look at the first page of the mid-unit 6 review to find the graph problems
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look at #13 on the test prep of 6.7 for the graph
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5. given that S[-2,1] f(x)dx = 4, S[1,5] f(x)dx = -3, and S[-2,1] g(x)dx = 8, find the following. a. S[5,1] f(x)dx b. S[-2,5] f(x)dx c. S[-2,1] (f(x) + 2g(x)]dx d. S[0,1] f(x)dx e. S[1,-2} 3f(x)dx f. S[5,5] [f(x) - g(x)]dx
a. 3 b. 1 c. 20 d. cannot be determined e. -12 f. 0
1. consider the region enclosed between the x-axis and the curve y = e^x a. use the left Riemann sum approximation with 5 equal subintervals to approximate the area of the region between x = -1 and x = 4. show your work. b. use a right Riemann sum approximation with 5 equal subintervals to approximate the same region
a. 31.56units b. 52.035 units
the area under the curve gives us the....
accumulation of change
width of the n subintervals
(b-a)/n interval: [a,b] n = number of subintervals
NEVER FORGET
+ c
Sine and Cosine integrals S sinxdx = S cosxdx =
-cosx + C sinx + C
find the indefinite integrals 1. S cos(2x + 1)dx 2. S 2xe^(3x^2)dx 3. S (6x^2 - 1)(2x^3 - x)^3dx
1. 1/2sin(2x + 1) + c 2. 1/3e^(3x^2) + c 3. (2x^3 - x)^4/4 + c
find the indefinite integral. 1. S (4x^2/(x-2))dx 2. S (28x^2 + 33x - 35)/(4x+7)dx 3. S (1/√(-x^2-4x-3))dx 4. S 8/(x^2 - 10x + 26)dx 5. S 6/(√-x^2 - 8x -7)dx 6. S (14x^2 + 11x - 5)/(2x+1)dx 7. S 1/(x^2 - 12x + 37)dx 8. S (5x^3 + 12x^2 - 38x - 54)/(5x+7) 9. S 3/(x^2 + 14x + 49)dx 10. S (x^3+2x^2+5)/(x+2)dx
1. 2x^2 + 8x + 16lnIx-2I + c 2. 7/2x^2 - 4x - 7/4lnI4x+7I + c 3. sin^(-1)(x+2) + c 4. 8 tan^(-1)(x-5) + c 5. 6 sin^(-1)(x+4/3) + c 6. 7/2x^2 + 2x - 7/2lnI2x+1I + c 7. tan^(-1)(x-6) + c 8. x^3/3 + x^2/2 - 9x + 9/5lnI5x+7I + C 9. -3/(x+7) + c 10. x^3/3 + 5lnIx+2I + c
let f be a continuous function that produces the following definite integral values. S[-6,-2] f(x)dx = -3 S[-6,4] f(x)dx = 2 1. S[-2,4] f(x)dx = 2. S[-2,-6] f(x)dx = 3. S[-6,4] 5f(x)dx =
1. 5 2. 3 3. 10
6.4 accumulation functions and the fundamental theorem of calculus (FTC) important ideas:
1. an accumulation function outputs the area under a curve from some starting value of x (the input) F(x) = S[c,x] f(t)dt FTC Part 1: Rate of change of an accumulation function d/dx S[c,x] f(t)dt = d/dx F(x) = f(x)
integrate. 1. S dx/√(16-x^2) 2. S dx/(25+4x^2) 3. S dx/(x√(x^2 - 16)) 4. S 2xdx/1+x^2 5. S x^2dx/1+x^2 6. (2x-5)/x^2+2x+2dx
1. sin^(-1)(x/4) + c 2. tan^(-1)(2x/5) + c 3. 1/4sec^(-1)(IxI/4) + c 4. lnIx^2 + 1I + c 5. x - tan^(-1)x + c 6. lnIx^2 + 2x + 2I - 7(tan^(-1)(x+1) + c)
using long division to rewrite the integrand: 1. S (3x^3 - x^2 - 5x + 1)/(x-2)dx 2. S (6x^4 - 7x^3 + x^2 + 2x)/(3x - 5)dx
1. x^3 + 5/2x^2 + 5x + 11lnIx-2I + c 2. x^4/2 + x^3/3 + x^2 + 4x + 20/3lnI3x - 5I + c
inverse trig derivatives: d/dx sin^(-1)x = d/dx sec^(-1)x = d/dx tan^(-1)x =
1/(√(1 - x^2)) 1/(IxI√(x^2 - 1)) 1/(x^2 + 1)
evaluate the definite integrals. 13. S[0,π/2] sin(2x)dx 14. S[-1/3,1/3] 1/(1+9t^2)dt 15. S[0,4] 1/√(2x+1)dx 16. S[π/4,0] tanxsec^2xdx 17. S[0,π/8]sec(2x)tan(2x)dx 18. S[1,e](lnx/x)dx 19. S[0,1] (x^2 + 2x)/(^3√x^3 + 3x^2 + 4) 20. S[0,π](2sinx + sin2x)dx
13. 1 14. π/6 15. 2 16. -1/2 17. (√(2) - 1)/2 18. 1/2 19. 2 - 1/2*(^3√16) 20. 4
find the function that satisfies the given conditions. 15. h'(t) = 8t^3 + 5 and h (1) = -4 16. dy/dx = 2x + sinx and y(0) = 4 17. f"(x) = x^(-3/2) and f'(4) = 2 and f(0) = 0 18. f"(x) = sinx and f'(0) = 1 and f(0) = 6
15. h(t) = 2t^4 + 5t - 11 16. y = x^2 - cosx + 5 17. f(x) = -4√x + 3x 18. f(x) = -sinx + 2x + 6
21. S[-1,1] 2/(1+x^2)dx = 22. S x√3xdx =
21. π 22. (2√3)/5x^(5/2) + c
using completing the square to rewrite the integrand: 3. S 1/(x^2 + 6x + 10)dx 4. S 1/(√-x^2 + 8x - 15)dx 5. S 32/(x^2 - 4x + 20)dx
3. tan^(-1)(x+3) + c 4. sin^(-1)(x-4) + c 5. 8 tan^(-1)(x-2/4) + c
3. S (1 - 5/x)dx 4. if f'(x) = 2/x + 6/x^2, find the particular solution of f(x) if f(1) = -7
3. x - 5lnIxI + c 4. f(x) = 2lnIxI - 6/x - 1
3. let f be the function given by f(x) = S[1/10,x] sin(1/t)dt for 1/10<x<1. at what value(s) of x does f attain a relative maximum? 4. let h be ht function given by h(x) = S[1,x] (1-e^cost)dt for 1<x<10. at what value(s) of x does h attain a relative min?
3. x = 0.159 4. x = 1.5707 and 7.8539
evaluate the definite integral. use a calculator to check. 4. S[-2,5] (4 - 6x)dx 5. S[1,4] (√x - 1/x^2)dx 6. S[0,π/2] 4sinxdx
4. -35 5. 47/12 6. 4
5. S(3sec^2x)dx 6. if dy/dx = 1 - sinx, find the particular solution of f(x) if f(π) = π - 2.
5. 3tanx + c 6. y = x + cosx - 1
use the given information to find the value of the function 6. if g'(x) = cosx and g(π) = 7, then g(3π/2) = 7. let h(x) be an antiderivative of x^2 - 2x. if h(-3) = 4, then h(1)
6. 6 7. 21.33
rewrite the definite integral using summation notation S[2,6] (x^2 - 3)dx =
lim n→∞ nEk=1 (4/n)((2 + 4k/n)^2 - 3)
area for a semicircle
(πr^2)/2
7. S (1/√(1-x^2))dx = 8. S (1/(IxI√(x^2 - 1)) =
sin^(-1)x + c sec^(-1)x + c
indefinite integrals and u-substitution find the indefinite integral: 1. S (3x-4)^5dx 2. S 6x^2(x^3 + 4)^5dx 3. S ((√x - 1)^2/(√x))dx 4. S sinxe^cosxdx 5. S cot(3x)dx need to solve for x: 6. S x/√(x+1)dx 7. S 1/√(1 - 4x^2)dx
((3x - 4)^6)/18 + c ((x^3 + 4)^6)/3 + c 2/3(√x -1 )^3 + c -e^cosx + c 1/3lnIsin3xI + c 2/3(x+1)^(3/2) - 2√x+1 +c 1/2sin^(-1)(2x) + C
10. S[-5,2] f(x)dx = -17 and S[5,2] f(x)dx = 4, what is the value of S[-5,5] f(x)dx?
-21
1. find the general solution of S 7^xdx 2. If f'(x) = e^x - 2x^2, find the particular solution of f(x) if f(0) = 4
1. (1/ln7)*7^x + C 2. f(x) = e^x - 2/3x^3 + 3
find the indefinite integrals: 1. S (x^2)/(1+x^3)^2dx 2. S cos√x/√xdx 3. S sinx/1+cos^2xdx 4. S 1/√(1-9x^2)dx 5. S e^xsine^xdx 6. S tanxcosxdx 7. S sec^2x/√tanx 8. S xdx/√(1-x^2) 9. S (lnx)^5/xdx 10. S 1/(25x^2 + 1) 11. S (2x + 5)(x^2 + 5x)^7dx 12. S e^x/(4 - e^x)dx
1. -1/(3 + 3x^3) + c 2. 2sin√x + c 3. -tan^(-1)(cosx) + c 4. 1/3sin^(-1)(3x) + c 5. -cos(e^x) + c 6. -cosx + c 7. 2√tanx + c 8. -√(1-x^2) + c 9. (lnx)^6/6 + c 10. 1/5tan^(-1)(5x) + c 11. (x^2 + 5x)^8/8 + c 12. -lnI4 - e^xI + c
find the indefinite integral: 1. S (3cscxcotx - 1)dx 2. S 3x(√x - x^2)dx 3. S 1/(√(-x^2 -10x - 24))dx 4. S 2^xdx 5. S 1/(x^2 - 4x + 5)dx 6. S (5-sec^2x)dx 7. S √x(x - 3/x)dx 8. S 1/√(1 - 9x^2)dx 9. S sec(5x)tan(5x)dx 10. S 4x^2/(x-2)dx 11. S (8/x - 1/x^2 - e^x)dx 12. S sinx/(1+cos^2x)dx 13. S (10x^2 - 24x + 12)/(5x - 2)dx
1. -3cscx - x + c 2. 6/5x^(5/6) - 3/4x^4 + c 3. sin^(-1)(x + 5) + c 4. 2^x/ln2 + c 5. tan^(-1)(x-2) + c 6. 5x - tanx + c 7. 2/5x^(5/2) - 6√x + c 8. 1/3sin^(-1)(3x) + c 9. 1/5sec(5x) + c 10. 2x^2 + 8x + 16lnIx - 2I + c 11. 8lnIxI + 1/x + e^x + c 12. -tan^(-1)(cosx) + c or cot^(-1)(cosx) + c 13. x^2 - 4x + 4lnI5x - 2I + c
find each indefinite integral. 1. S 5sin5xdx 2. S 2sec^2(2x-1)dx 3. S (8x+1)^3dx 4. S x^2√x^3 5. S x/(1-x^2)^3dx 6. S 1/√2xdx 7. S x√(x+4)dx 8. S (2x-1)/(x^2-x)^3dx
1. -cos5x + c 2. tan(2x-1) + c 3. (8x-1)^4/32 + c 4. 2/9√(x^3 + 1)^3 + c 5. 1/(4(1-x^2)^2) + c 6. √(2x) + c 7. 2/5(x+4)^(5/2) - (8/3)(x+4)^(3/2) + c 8. -1/(2√x^2 -x) + c
find the value of the indefinite integral. 1. S[-2,-1] (1/x^2 + x^2 - 5x)dx 2. S[-1,8] (x^(2/3) - x)dx 3. S[0,π] (x - sinx)dx 4. S[-1,1] x√(1-x^2)dx 5. S[0,π/2] (sin(2x))/(cos^2(2x))dx 6. S[e,e^2] 1/xlnxdx
1. 10.33 2. -11.7 3. (π)^2/2 - 2 4. 0 5. 1/2 6. 0.693
find the following indefinite integrals 1. S (2x^2 - 3/x + 2^x)dx 2. S((x^7 - 2x)/x^2)dx 3. S √x(x - ^4√x)dx
1. 2/3x^3 - 3lnIxI + 2^x/ln2 + c 2. x^6/6 - 2lnIxI + c 3. 2/5x^(5/2) - 4/7x^(7/4) + c
the rate at which water is being pumped into a tank is given by the continuous and increasing function R(t). a table of selected values of R(t), for the time interval 0<t<12 minutes, is given below. Time (minutes): 0, 3, 6, 9, 12 R(t) (gallons/min): 7, 13, 18, 23, 27 use the following Riemann Sums (with the given intervals), to estimate the number of gallons of water pumped into the tank during the 12 minutes. 1. Right-Riemann Sum with 4 subintervals. Is the approximation greater or less than the true value? Why? 2. Left-Riemann Sum with 4 subintervals. Is the approximation greater or less than the true value? Why? 3. Midpoint-Riemann Sum with 2 subintervals. 4. Trapezoidal Sum with 4 subintervals.
1. 243 gallons; greater than bc it is a right Riemann sum on an increasing function 2. 183 gallons; less than bc it is a left Riemann sum on an increasing function 3. 216 gallons 4. 213 gallons
find the indefinite integral. 1. S (6x^2)/(x+1)dx 2. S 1/(x^2 + 6x + 10)dx
1. 3x^2 - 6x + 6lnIx+1I + c 2. tan^(-1)(x+3) + C
find the following indefinite integrals. 1. S(6^x - 1/x)dx 2. S((x^5 - 6)/x)dx 3. S(e^x + e^2)dx 4. S 5^xdx 5. S (1/(x*^3√x)dx 6. S(3-x)^2dx 7. S√t(t - 1/t)dt 8. S((5x^2 + x - 2)/x)dx 9. S(x - 2csc^x)dx 10. S(x^2 + 2)^2dx 11. S(3cscxcotx - 1)dx 12. S((√x - x - 5)/x)dx 13. S(5-sec^2x)dx 14. S(3sinx - √x)dx
1. 6^x/ln6 - lnIxI + c 2. 1/5x^5 - 6lnIxI + c 3. e^x + e^(2)x + c 4. 5^x/ln5 + c 5. -3x^(-1/3) + c 6. 9x - 3x^2 + 1/3x^3 + c 7. 2.5t^(5/2) - 2t^(1/2) + c 8. 5/2x^2 + x - 2lnIxI + c 9. 1/2x^2 + 2cotx + c 10. 1/5x^5 + 4/3x^3 + 4x + c 11. -3cscx - x + C 12. 2√x - x - 5lnIxI + c 13. 5x - tanx + c 14. -3cosx - 2/3^(3/2) + c
Find F'(x) 1. F(x) = S[4,x] (1/√t)dt 2. F(x) = S[3,x] t^2dt 3. F(x) = S[π,x] tantdt 4. F(x) = S[5,x] (1/t)dt 5. F(x) = S[-1,2x] (1-t^2)dt 6. F(x) = S[e,e^x] lntdt 7. F(x) = S[9,x^4] (√t)dt
1. F'(x) = 1/√x 2. F'(x) = x^2 3. F'(x) = tanx 4. F'(x) = 1/x 5. F'(x) = 8x - 32x^3 6. F'(x) = e^xlne^x 7. F'(x) = 4x^5
find the indefinite integral and check by integration 1. S (2x - 3x^2)dx 2. S (√x + 1/2√x)dx 3. S ^3√x^2dx 4. S (x^2+2x-3)/x^4dx 5. S xdx 6. S (t^2 - cost)dt 7. S (sec^2x - sinx)dx 8. S secx(tanx-secx)dx 9. S 2cos(2x)dx 10. S S (x^2 + 1)^2(2x)dx
1. x^2 - x^3 + c 2. 2/3√x^3 + √x + c 3. 3/5x^(5/3) + c 4. -1/x - 1/x^2 + 1/x^3 + c 5. x^2/2 + c 6. t^3/3 - sint + c 7. tanx + cosx + c 8. secx - tanx + c 9. sin(2x) + c 10. (x^1 + 1)^3/3 + c
using a calculator to find an integral value: sketch a graph of the definite integral. use the calculator to evaluate. 10. S[2,3] √(x-1)dx 11. S[-2,4] (x/3 - 1)dx 2. S[0,3] -√(x+1)dx 3. S[-2,3] Ix+1Idx 4. S[1,-3] (-x/2 + 1)dx
10. 1.219 11. -4 2. -4.667 3. 8.500 4. -6
11. S 16/(x^2 - 6x + 25)dx 12. S 12/(√-x^2 - 2x + 3)dx
11. 4tan^(-1)(x-3/4) + c 12. 12 sin^(-1)(x+1/2) + c
find the following indefinite integrals. 11. S (x^2 - x + 5)/xdx 12. S secxtanxdx 13. S (e^x + 2^x)dx 14. S (1/x + 1/x^3)dx 15. S √x(x - 4/x)dx 16. S (50x^3 - 55x^2 - 26x + 33)/(10x - 7)dx 17. S 1/(x^2 + 2x + 2)dx 18. If f'(x) = sin(e^x) and f(0) = 5.7, then f(2) =
11. x^2/2 - x + 5lnIxI + c 12. secx + x 13. e^x + 2^x/ln2 + c 14. lnIxI - x/2x^2 + c 15. 2/5x^(5/2) - 8√x + c 16. 5/3x^3 - x^2 - 4x + 5lnI10x - 7I + c 17. tan^(-1)(x + 1) + c 18. 6.251
evaluate the definite integral 14. S[1,4] (1/√x - x^2)dx 15. S[-1,2] (3x^2 - 4/x^2 + 1)dx 16. S[0,π] (sinx - 1)dx 17. S[4,16] -√xdx 18. S[1,2] e^(1-x)dx 19. S[0,1] 2x/√(x^2 + 1)dx 20. S[0,π/8] tan(2x)sec^2(2x)dx
14. -19 15. 18 16. 2 - π 17. -112/3 18. 1 - 1/e 19. 2√2 - 2 20. 1/4
14. let f(x) = S[0,x^2] costdt. at how many points in the closed interval [-√π,√π] does the instantaneous rate of change of f equal the average rate of change of f on that interval? 15. given h(x) = {x - 1 for x<0; sinx for x≥0, find S[-1,π] f(x)dx 16. a cubic polynomial function f is defined by f(x) = 2/3x^3 + ax^2 + bx + c, where a, b, and c are constants. the function f has a local minimum at x = -2, and the graph of f has a point of inflection at x = -5. if S[0,1] f(x)dx = 15/2 what is the value of c?
14. three 15. 1/2 16. c = -12
5. use a trapezoidal approximation with 4 subintervals to approximate the area under f(x) = 1/4x^2 - 2x + 6 on [-3,0]
29.320 units
separating rational functions: S (3x^2 + x - 2)/x dx
3/2x^2 + x - 2lnIxI + c
4. use a left Riemann sum with 4 subintervals to approximate the integral based of the values in the table. S[0,10] f(x)dx x: 0,4,6,7,10 f(x): 3,2,4,5,7
35 units
let g(x) = d/dxS[0,x] √(t^2 + 9)dt. what is g(-4)?
5
let f be a continuous function that produces the following definite integral values. S[-4,6] f(x)dx = 2 S{[6,8] f(x)dx = -5 find the following 5. S[-4,6] 5f(x)dx = 6. S[-4,8] f(x)dx = 7. S[8,6] f(x)dx =
5. 10 6. -3 7. 5
combining expressions: S 3x(√x + x^2)dx
6/5x^(5/2) + 3/4x^4 + c
evaluate the definite integral. 8. S[0,2] t^2√(t^3 + 1)dt 9. S[0,π/2] cosx√sinxdx`
8. 52/9 9. 2/3
derivative: d/dxn = 0 d/dxx = 1 d/dxx^n = nx^(n-1) d/dxe^x = e^x d/dxlnx = 1/x d/dxn^x = n^xlnx d/dxsinx = cosx d/dxcosx = -sinx d/dxtanx = sec^2x d/dxcotx = -csc^2x d/dxsecx = secxtanx d/dxcscx = -cscxcotx d/dxarcsinx = 1/√1-x^2 d/dxarccosx = -1/√1-x^2 d/dxarctanx = 1/1+x^2 d/dxarccotx = -1/1+x^2 d/dxarcsecx = 1/IxI√x^2-1 d/dxarccscx = -1/IxI√x^2-1 integral (antiderivative):
S 0dx = C S 1dx = x + C S x^n = (x^(n+1))/(n+1) + C S e^xdx = e^x + c S 1/xdx = lnx + c S n^xdx = n^x/lnn + c S cosxdx = sinx + c S sinxdx = -cosx + C S sec^2xdx = tanx + c S csc^2xdx = -cotx + c S tanxsecxdx = secx + c S cotxcscxdx = -cscx + c S 1/√1-x^2dx = arcsinx + c S -1/√1-x^2 = arccosx + c S 1/1+x^2 = arctanx + c S -1/1+x^2 = arccotx + c S 1/IxI√x^2-1 = arcsecx + c S -1/IxI√x^2-1 = arccscx + c
the expression 1/10(cos(1/10) + cos(2/10) + ... cos(10/10)) is a Riemann Sum approximation for what definite integral? where is the 10? why isn't it written in the integral?
S[0,1] cos(x)dx the 10 represents how many subintervals there are. it has nothing to do with the function. if it had been a 50 instead of a 10, it would have been a more accurate approximation.
what is the difference between IS[a,b] f(x)dxI and S[a,b] If(x)Idx?
When the absolute value sign is on the outside, you need to figure out the integral, then take the absolute value of it For the absolute value sign on the inside, no areas are negative (which means that the absolute value is taken before adding)
look at #2 and #3, and #1-#3 on the notes and practice in 6.1 on pages 1-3 to find areas of graphs
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look at #19 and #20 on the test prep of 6.4 for the graph
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look at page 2 of 6.5 for graph problems
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look of unit 6 review 6.1 to 6.5 for #1 on the first page for the graph
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look at the first (third) page of unit review part 2 for the table and graph
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the rate of field consumption, in gallons per minute, recorded during an airplane flight is given by the twice-differentiable and strictly increasing function R of time t. A table of selected values of R(t) for the time interval 0≤t≤90 minutes is shown below. At t=0 the plane had already consumed 84 gallons of fuel. time (minutes): 0,30,40,50,70,90 R(t) (gallons per minute): 20, 30, 40, 55, 65, 70 a. use data from the table to find an approximation for R'(45). show the computations that led to your answer. indicates units of measure. b. using a trapezoidal approximation with five subintervals, approximate how much field the plane has consumed after 90 minutes.
a. 1.5 gallons/min^2 b. 4,209 gallons
exponential: S (e^x)dx = S (a^x)dx =
e^x + C (1/lna)*a^x + C
properties of definite integrals:
equivalent limits: S[a,a] f(x)dx = 0 reversal of limits: S[b,a] f(x)dx = -S[a,b] f(x)dx multiply by constant (k = constant): S[a,b] kf(x)dx = kS[a,b] f(x)dx adjacent intervals (a<c<b): S[a,c] f(x)dx + S[c,b] f(x)dx = S[a,b] f(x)dx addition: S[a,b] [f(x) + g(x)]dx = S[a,b] f(x)dx + S[a,b] g(x)dx subtraction: S[a,b] [f(x) - g(x)]dx = S[a,b] f(x)dx - S[a,b] g(x)dx
find the derivative of each function f(x) = x^2 + 1 f(x) = x^2 - 5 f(x) = x^2 + 1000
f'(x) = 2x 2x 2x
how to find antiderivative:
f(x) = x^n F(x) = (x^(n+1))/(n+1) step one: add one to the exponent step two: divide by the new exponent *this is the reverse/opposite order of the power rule for finding a derivative*
Midpoint-Riemann Sum
find each midpoint of the subintervals to evaluate the integrals (estimate area)
steps of u-substitution
find the inner part and set it equal to u take the derivative of both sides to find what dx = plug u and whatever dx = into equation and take antiderivative plug inner part back into u and add c
integrals without a boundary are called __________. in the past, we've been working with definite integrals where the boundaries are given for the area under the curve. now we will work with indefinite integrals, which means we are finding an antiderivative as a general solution
indefinite integrals
the following problems are definite integrals, but use strategies that were covered in lesson 6.8: 19. S[1,3] (x+6)/x^2 = 20. S[-1,1] (4/1+x^2)dx =
ln3 + 4 2π
logarithm as the answer: S (1/x)dx =
lnIxI + C
techniques for differentiation learned in unit 6
power rule (antiderivatives) u-substitution trig inverse trig exponential and logarithms long division completion the square integration by parts linear partial fractions improper integrals
S 2cos2xdx S sinxcosxdx S x^2√(x^3 - 1)dx S x√(1-x)dx
sin2x + c 1/2sin^2x + c 2/9√(x^3 -1)^3 + c 2/5(1-x)^(5/2) - 2/3(1-x)^(3/2) + c
three new integral forms: S du/(√(a^2 - u^2)) = S du/(a^2 + u^2) = S du/(u√(u^2 - a^2)) =
sin^(-1)(u/a) + c (1/a)*tan^(-1)(u/a) + c (1/a)*sec^(-1)(IuI/a) + c a is a constant u is a function of x
trig integrals S cosxdx = S sinxdx = S sec^xdx = S cscxcotxdx = S secxtanxdx = S csc^2xdx =
sinx + C -cosx + c tanx + c -cscx + c secx + c -cotx + c
the sum of the area of all rectangles gives you the area under the curve
trianglex1*f(x1) + traingelx2*f(x2) + ... trainglexn*f(xn)
if n→∞ on the interval [a,b], what does the width of each subinterval (rectangle) approach?
trianglex→0
formulas for derivatives of inverse trig functions: 1. d/dx(sin^(-1)u) = 2. d/dx(cos^(-1)u) = 3. d/dx(tan^(-1)u) = 4. d/dx(cot^(-1)u) = 5. d/dx(sec^(-1)u) = 6. d/dx(csc^(-1)u) =
u'/√(1-u^2) -u'/√(1-u^2) u'/(1+u^2) -u'/(1+u^2) u'/(IuI√(u^2-1)) -u'/(IuI√(u^2-1))
remember the chain rule? we have something similar when we take antiderivatives. it is called ___________. look at the integrand (f(x) part of S f(x)dx) and think about taking the derivative. if the derivative requires the chain rule, then the antiderivative will require u-substitution
u-substitution
when the function is increasing, the Left-Riemann Sum is a... the Right is a...
underestimate overestimate
Left Riemann Sum
use rectangles with left-endpoints to evaluate integral (estimate area)
Right Riemann Sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
you know if you selected the correct inner part for u to be equal to if the...
variables or denominator and other numerator cancel out if you got dx correct
behavior of accumulation function F(x) is/has... increasing decreasing relative max relative min concave up concave down a point of inflection ...when or...
when: F'(x)>0 F'(x)<0 F'(x) changes from + to - F'(x) changes from - to + F"(x)>0 F"(x)<0 F"(x) changes sign or: f(x)>0 f(x)<0 f(x) changes sign from + to - f(x) changes sign from - to + f'(x) >0 f'(x)<0 f'(x) changes sign
if n is the number of subintervals (rectangles) on the interval [a,b], what is the width of each subinterval (rectangle)?
width = (triangle(which means difference/change))x = (b-a)/n
S x^4/4+x^2dx S dx/(x^2 + 4x + 13)dx
x^3/3 - 4x + 8tan^(-1)(x/2) + c 1/3tan^(-1)((x+2)/3) + c
look at the domain and range of inverse trig functions on the three new integral forms
yUH!!!!!!!!!!
to find the area under the curve....
you find the area of each normal shape you can find the area of and then add them up
look at #3 and #4 on page 1 of 6.6 for the graphs
yuh000000
look at the top of the first page of 6.6 for the graph problem
yuhhhuhuhuhuuhu
21. if S[0,k] x/(x^2 + 6)dx = 1/2ln6, where k>0, then k = 22. the function f is continuous and S[4,19] f(u)du = 10. what is the value of S[1,4][xf(x^2 + 3)]dx 23. S 1/(√(16 - x^2))dx
√30 5 sin^(-1)(x/4) + c
the approximation method is called a...
Reimann Sum. it was named after a German mathematician named Bernhard Riemann
lim n→∞ 2/n((1/((1/n) + 3)) + (1/((2/n) + 3)) + (1/((3/n) + 3)) + (1/((2n/n) + 3)) assuming the lower limit "a" is a 0, write a definite integral that represents the above expression
S[0,2] (1/(x+3))dx
the expression 3/7(3/ysin(3/7) + 6/7sin(6/7) + ... 21/7sin(21/7)) is a Riemann sum approximation of which of the following integrals?
S[0,3] xsinxdx
write a definite integral that is equivalent to the given summation notation. the lower limit for the integral is also given to help you get started. 1. integral's lower limit = 0 lim n→∞ nEk=1 (π/n)(cos(πk/n)) 2. integral's lower limit = -3 lim n→∞ nEk=1 (5/n)(^3√(-3 + 5k/n)) 3. integral's lower limit = 6 lim n→∞ nEk=1 (9/n)(1/(6 + 9k/n)^2)
S[0,π] cosxdx S[-3,2] ^3√xdx S[6,15] 1/x^2dx
which of the following expressions is equal to lim n→∞ 1/n(e^(1 + 1/n) + e^(1 + 2/n) ... + e^(1 + n/n))
S[1,2] e^xdx
the expression 1/5(ln(2 + 1/5) + ln(2 + 2/5) + ... + ln(2 + 5/5)) is a Riemann sum approximation of which of the following integrals?
S[2,3] lnxdx
look at #5 on the practice and all of the test prep in 6.5 for the table and graphs
YUH!!!!!!!
look at #1 on page 1 of 6.4 for the graph
YUH%%%%% this is called an accumulation function
4. let y(t) represent the weight loss per week of a contestant on the Biggest Loser, where y is a differentiable function of t. the table shows the weight loss per week recorded at selected times. time (week): 2, 4, 7, 8, 11 y(t) (pounds/week): 14, 12, 18, 14, 17 a. use the data from the table and a left Riemann sum with four subintervals. show the computation that lead to your answer. b. what does your answer represent in this situation?
a. 124 pounds b. a contestant on the Biggest Loser has lost 124 pounds over the 9 week period
9. let y(t) represent the rate of change of the population of a town over a 20-year period, where y is a differentiable increasing function of t. the table shows the population change in people per year recorded at selected times. time (years): 0,4,10,13,20 y(t) people per year: 2500,2724,3108,3697,4283 a. use the data from the table and a right Riemann Sum with four subintervals to approximate the area under the curve. b. what does your answer from part (a) represent? c. assuming that y(t) is a continuous increasing function, is your approximation from part (a) greater or less than the true value? why?
a. 70,616 people b. the town has 70,616 more people after the 20-year period. c. greater than the true value because it is a right Riemann sum on an increasing function
rewrite the summation notation expression as a definite integral (3 times). lim n→∞ nEk=1 (6/n)(4 + 6k/n)^2 =
a. S[0,6] (4 + x)^2dx b. S[4,10] x^2dx c. S[0,1] 6(4 + 6x)^2dx
10. a rectangular pool gets deeper from one end of the pool to the other. the table shows the depth h(x) of the water at 4-foot intervals from one end of the pool to the other. position, x (feet): 0,4,8,12,16,20,24,28,32 h(x) feet: 6.5,8,9.5,10,11,11.5,12,13,13.5 a. use the data from the table to find an approximation for h'(10), and explain the meaning of h'(10) in terms of the depth of the pool. show the computations that lead to your answer. b. use a midpoint Riemann Sum with 4 subintervals to approximate the area under the curve.
a. at 10 feet from one side of the pool, the depth is changing by 0.125ft for every foot from one side of the pool. b. 340 feet
when we take an integral, it is taking the...
antiderivative of a function. the area under the curve is represented by an antiderivative
Fundamental Theorem of Calculus
c = a in the picture if a is a constant and f is a continuous function, then... (picture insert here)
an antiderivative of a function f(x) is a...
function F(x) whose derivative if f(x) for example, let f(x) = 3x^2 and f'(x) = 6x. the expression 3x^2 is an antiderivative of 6x
variations of the FTC
if a is a constant, f is a continuous function, and g and h are differentiable then...
the fundamental theorem of calculus (part 1)
if f is continuous on the interval [a,b], then the area under the curve of f from [a,b] can be represented by S[a,b] f(x)dx = F(b) - F(a) where F(x) is the antiderivative of f
derivatives and integrals are _______ of each other. they cancel each other out, just like multiplication and division. in example 1, the graph of f is the derivative of F(x). so F(x) is considered the __________ of f(x).
inverses antiderivative
if you are trying to estimate the area using Riemann sums and an equation,...
just plug in the left, right, midpoint, or trapezoid points of the x-axis to find the height (y-axis)
write a summation notation equivalent to the definite integral. 4. S[-3,3] x^2dx 5. S{2,5} 1/xdx 6. S[0.7] √xdx
lim n→∞ nEk=1 (6/n)(-3 + 6k/n)^2 lim n→∞ nEk=1 (3/n)(1/(2 + 3k/n)) lim n→∞ nEk=1 (7/n)(√7k/n)
Trapezoidal Sum
multiply the width by the (left and right ends of the subinterval that create a trapezoid divided by two) and then add them all together depending not he number of subintervals
Left, Right, and Midpoint Riemann Sum
multiply the width by the height (whether left, right, or midpoint) and then find the sum of each depending on the number of subintervals
let us say we know where the interval starts at a, but we do not know where it stops. that would give us [a,x] where is a constant and x is some unknown variable. we can represent that as a new function that looks like this:
F(x) = S[a,x] f(t)dt
Accumulation Function
F(x) = S[a,x] f(t)dt a is a constant F'(x) = f(x)
which of the following definite integrals are equal to lim n→∞ nEk=1 (-1 + 4k/n)^2(4/n) 1. S[-1,3] x^2dx 2. S[0,4] (-1 + x)^2dx 3. S[0,1] (4(-1 + 4x)^2)dx
1, 2, and 3
approximate the area under the curve using the given Riemann Sum 1. f(x) = 1/5x^3 - x + 7; midpoint Riemann sum on the interval [-1,2] with n = 3 subintervals 2. f(x) = 6x + 5; left Riemann sum on [-2,2] with n = 5 subintervals
1. 20.175 units 2. 13 units
find the value of the definite integral. use a calculator to check. 1. S[0,4] (2x + 4)dx 2. S[0,π/2] (sinx - x)dx 3. S[-1,3] (6x^2 - 8)dx 4. S[4,9] (1/√x)dx 5. S[-4,-1] (3/x^2 + 1)dx 6. S[-π/2,0] (2-cosx)dx
1. 32 2. 1 - π^2/8 3. 24 4. 2 5. 5 1/4 6. π - 1
find F'(x). 1. F(x) = S[2,x] (3t^2 + 4t)dt 2. F(x) = S[π/2,x] sin(t)dt 3. F(x) = S[1,4x] h(t)dt 4. F(x) = S[-x,x] 5tdt 5. F(x) = S[2x,3x] (t^2 - t)dt 1. F(x) = S[2,x] t^3dt 2. F(x) = S[0,x] 5dt 3. F(x) = S[-1,x] (4t - t^2)dt 4. F(x) = S[π,x] cos(t)dt 5. F(x) = S[1,x^2] t^3dt 6. F(x) = S[π,x^2] sin(t)dt 7. F(x) = S[π, sinx] 1/tdt 8. F(x) = S[4,x^2] 3√tdt 9. F(x) = S[0,3x] 2tdt 10. F(x) = S[0,tanx] t^2dt 11. F(x) = S[3,x^2] tan(t)dt 12. F(x) = S[3,g(x)] sec(t)dt 13. F(x) = S[1,2x] f(t)dt 14. F(x) = S[x, x+2] (4t + 1)dt 15. F(x) = S[-x^2,x] (3t - 1)dt 16. F(x) = S[-x,x] t^3dt 17. F(x) = S[2x,3x] t^2dt
1. F'(x) = 3x^2 + 4x 2. F'(x) = 3x^2sin(x^3) 3. F'(x) = 4h(4x) 4. F'(x) = 0 5. F'(x) = 19x^2 - 5x 1. F'(x) = x^3 2. F'(x) = 5 3. F'(x) = 4x - x^2 4. F'(x) = cosx 5. F'(x) = 2x^7 6. F'(x) = 2xsin(x^2) 7. F'(x) = cotx 8. F'(x) = 6x^2 9. F'(x) = 18x 10. F'(x) = tan^2xsec^2x 11. F'(x) = 2xtan(x^2) 12. F'(x) = sec(g(x)*g'(x)) 13. F'(x) = 2f(2x) 14. F'(x) = 8 15. F'(x) = -6x^3 + x - 1 16. F'(x) = 0 17. F'(x) = 19x^2
given the function f(x), find the antiderivative F(x). 1. f(x) = x^2 2. f(x) = 5x^3 + 6/x^3 - 1 3. f(x) = √x + (3/√x)
1. F(x) = x^3/3 + C 2. F(x) = 5/4x^4 - 3/x^2 - x + C 3. F(x) = 2/3x^(3/2) + 6√x + C
1. snow starts falling at 6am. the rate that snow is falling overnight is represents by the equation S'(t) where S'(t) is measured in inches per hour, and t represents hours since midnight. write an integral expression that gives the total amount of snow that falls during the first 3 hours of snow fall. 2. the function C'(t) gives the rate, in bars per minute, at which chocolate bars are being removed from a shelf at the grocery store, where t is measured in minutes. using the correct units, interpret the meaning of the following expression in the context of this problem. S[0,9] C'(t)dt = 46 3. Paul is using rectangles to find the area under the curve on the interval [-2,7]. use integral notation to express the following limit. lim n(arrow)∞ nEi=1 (3xi^2 - 5xi)(trianglex)
1. S[0,3] S'(t)dt 2. over the course of 9 minutes, the total number of chocolate bars that are removed from a shelf at the grocery store is 46 bars. 3. S[-2,7] (3x^2 - 5x)dx 4.YUH!!!!!!
write a definite integral that is equivalent to the given summation notation. the lower limit for the integral is also given to help you get started. 1. integral's lower limit = 0 lim n→∞ k=1En (π/4n)tan(πk/4n) 2. integral's lower limit = -1 lim n→∞ k=1En(n is on top) (8/n)[4(-1 + 8k/n)] write a summation notation equivalent to the definite integral. 3. S[-1,3] x^2dx 4. S[3,4] lnxdx
1. S[0,π/4] tanxdx 2. S[-1,7] 4xdx 3. lim n→∞ k=1En (4/n)(-1 + 4k/n)^2 4. lim n→∞ k=1En (1/n)(ln(3 + k/n))
6.1 exploring accumulation of change important ideas:
1. the accumulation of a quantity is represented by the area underneath its derivative 2. accumulated quantity = Rate1*(triangle)t1 + Rate2*(triangle)t2 + ...... + Raten*(triangle (which means rate of change))tn 3. units for area underneath a rate of change curve; units of rate of change * units of independent variable 4. area above x-axis is positive area→positive accumulation→quantity is increasing 5. area below x-axis is negative area→negative accumulation→quantity is decreasing
6.2 approximating areas with Riemann Sum important ideas:
1. to approximate the area on [a,b] with equal subdivisions use (b-a)/n as the width of each rectangle 2. to find the height of each rectangle: a. left endpoint - evaluate function at left endpoint of each interval b. right endpoint - evaluate function at right endpoint of each interval c. middle - evaluate function at midpoint of each interval 3. total area/accumulation = ROCx1*f(x1) + ..... + ROCxn*f(xn) 4. if a function is increasing, the left endpoint gives an underestimate, and the right endpoint gives an overestimate 5. if a function is decreasing, the left endpoint gives an overestimate, and the right endpoint gives an underestimate
6.3 Riemann Sums, summation notation, and definite integrals important ideas:
1. to find the exact are under a curve f(x) over the interval [a,b] use infinitely many rectangles of infinitesimally small widths. integral notation vs summation notation
let f and g be continuous functions that produce the following definite integral values. S[1,2] f(x)dx = -2 S[1,6] f(x)dx = 4 S[1,6] g(x)dx = 8 Find the following: 12. S[2,2] g(x)dx 13. S[6,1] g(x)dx 14. 3S[1,2] f(x)dx 15. S[2,6] f(x)dx 16. S[1,6] [f(x) - g(x)]dx 17. S[1,6] [3f(x) - g(x)]dx 18. S[1,6] If(x) - g(x)Idx 19. IS[1,6] f(x) - g(x)dxI
12. 0 13. -4 14. -6 15. 6 16. -4 17. 4 18. cannot be determined 19. 4
7. If f'(x) = x^2 - 2 and f(1) = -2, then f(3) =
2 2/3
you are on a road trip and have your car on cruise control for 4 hours. you travel at 60mph. how far have you traveled
240 miles
each function listed represents a rate of change. what are the units for the area under the curve? 4. f(t) is measured in milligrams per year and t is measured in years. 5. g(t) is measured in gallons per month and t is measured in months. 6. h(t) is measured in feet per hour and t is measured in hours.
4. milligrams 5. gallons 6. feet
let f and g be continuous functions that produce the following definite integral values. S[-3,2] f(x)dx = 2 S[2,7] f(x)dx = -5 S[-3,2] g(x)dx = 6 find the following: 5. S[2,7] 2f(x)dx 6. 4S[-3,2] f(x)dx 7. S[-3,7] f(x)dx 8. S[2,-3] g(x)dx 9. S[-3,2] [g(x) - f(x)]dx 10. IS[2,7] f(x)dxI 11. -S[7,2] f(x)dx
5. -10 6. 8 7. -3 8. -6 9. 4 10. 5 11. -5
piecewise-functions and integrals: 6. the function g is defined by g(x) = {3 for x<2; 4-x for x≥2. what is the value of S[1,5] g(x)dx? 7. what is the value of S[0,5] Ix - 2Idx?
6. 4.5 7. 6.5
11. the rate at which customers are being served at Starbucks is given by the continuous function R(t). a table of selected values of R(t), for the time interval 0<t<10hours is given below. at t=10 there had already been 200 customers served. time (hours): 0,2,3,6,10 R(t) (people/hr): 37,44,36,42,48
618 customers
find F'(x) 7. F(x) = S[0,cosx] t^2dt 8. F(x) = S[x^2,8-x] (2t + 5)dt
7. -sinxcos^2x 8. -4x^3 - 8x - 21
Use the given information to find the value of the function. 7. If f'(x) = cosx and f(-π) = 12, then f(3π/2) = 8. If f'(x) = sin(3x) + e^x and f(1) = 0.751, then f(4) = 9. let f be a differentiable function such that f(1) = 4 and f'(x) = 6x^2 + 3. what is the value of f(3)? 10. let f be a differentiable function such that f(0) = -0.5 and f'(x) = 2 - cos(ex). what is the value of f(-2)? 11. let h(x) be an antiderivative of 5 - 3x. if h(-1) = -3, then h(2) = 12. Let F(x) be an antiderivative of lnx/x. if F(2) = -0.13, then F(5) =
7. 11 8. 52.020 9. 62 10. -4.776 11. 7.5 12. 0.925
when the function is concave up, the trapezoidal estimation is a... when the function is concave down, the trapezoidal estimation is a...
overestimate underestimate
when the function is decreasing, the Left-Riemann Sum is a... the Right is a...
overestimate underestimate
if it is above the x-axis, the area is... if it is below the x-axis, the area is...
positive negative
on the example of g(x) = S[a,x] f(t)dt where the graph of f is shown below and a is a constant...
relative min of g: crosses x-axis from neg to pos relative max of g: crosses x-axis from pos to neg interval were g is concave up: increasing (slope on graph is going up) intervals were g is concave down: decreasing (slope on graph is going down) point of inflection(s) of g: corner, f'(x) = 0, top/bottom of circle, f'(x) change of sign
definite integral notation
the area under the cure of f(x) on the interval [a,b]:
summation notation
the area under the curve of f(x) on the interval [a,b] is represented by lim n→∞ (insert picture here) (b-a / n)*f(a + (b-a / n)*k) b-a/n is the width (a + (b-a / n)*k) is the height I = k n represents the number of subinterval (rectangles) there are int he interval [a,b] and k represents the 4th subinterval or you can write it like this: lim trianglex→0 (insert picture with i = k) (trianglex)(f(x(subscript)k))
units for area under the curve:
the dependent unit multiplied by the independent unit. in other words, the unit for y times the unit for x.
2. given s(t) represents inches of snow per hour and t represents hours, what does the area under the curve of s(t) represent?
the number of inches of snow lost or gained within the time period
area under the curve
the region between a function and the x-axis is called the area under the curve. "under" in this instances does not mean below. it means between the x-axis and the function. looks at graphs for example