Unit 8
trapezoidal rule
(or h/2[first base + second base])
how to find volume of solid with cross sections
1. find formula for area of the shape of the cross section given 2. using the given equation, solve for x or y, depending on if the cross sections are perpendicular to a certain axis 3. look at the graph. the cross section may be split into 2 x's or y's, meaning that b will equal 2y or 2x. you will have to multiply the integral by if thats the case. 4. if cross sections are isosceles right trangles w/ hypotenuse as leg, A = y² 5. plug in the solved equation from step 2 into the area formula. 6. integrate according to interval
general approach for finding area between two curves (*X*)
1. first, find where the two curves intersect. this will establish the interval 2. sketch, if graph is not given. use easy points 3. solve integral from a to b of upper function - lower function dx
the average value of y = k(x) equals 4 for [1, 6], and equals 5 for [6, 8] find the average value of k(x) for [1,8]
1. set the average value of k(x) from 1 to 6 equal to 4 2. solve for the first x (should be 20) 3. set the average value of k(x) from 6 to 8 equal to 5 4. solve for the second x (should be 10) 5. solve: (1/7)(20+10) = 30/7
total distance traveled on [a, b]
A + B + C
displacement on [a, b]
A - B + C
area of isosceles triangle with base as leg
A = 1/2b²
area of rectangle with h = 1/2b
A = 1/2b²
area of isosceles triangle with hypotenuse as leg
A = 1/4b²
area of equilateral triangle
A = √3/4b²
area of rectangle
A=bh
area of square
A=s²
area of semicircle
A=½πr² if b = diameter, then r = b/2, so area can also be: A=π/8b²
how to solve the following questions: you drop your calculus book from a window 48 ft above the ground. a) how long will it take your book to reach the ground b) what is the velocity of the book at the instant it reaches the ground
a) we know that the acceleration of gravity in feet is a(t) = -32 ft/seconds squared. so, v(t) would be equal to the integral of that. when you integrate, v(t) should equal -32t + C1. you would then integrate once more to find the position function s(t), which is -16t^2 + C2. we know that the original position/condition was 48, so s(t) would really be s(t) = -16t^2 + 48. solve for t when s(t) = 0. t should be square root 3. b) plug in square root 3 into the velocity function
if a function is integrable on an interval, the function is also....
continuous on that interval
how do you distinguish between the disc and washer method?
disc method = function touches axis of revolution washer method = function does not touch
average velocity
displacement/time
actual position
initial condition is added to integral
if graph is symmetric...
multiply the integral from 0 to b by 2 since both sides are symmetric
how to find a number c that satisfies the conclusion of the MVT for integrals
once you've found the average value, set the function being integrated equal to that average value and solve for x (which would be c)
when revolving around the y-axis, you must tilt you head to the...
right
finding area between curves expressed as functions of *Y*
same approach as when you're finding area with x-axis approach, except we're doing it with y-axis
finding volume when revolving around other axes
same procedure as when you're revolving around the y or x-axis, except... to rotate around a *horizontal line*, y = c, express outer radius and inner radius as *functions of x* and integrate along the *x-axis* to rotate around a *vertical line*, x = c, express outer radius and inner radius as *functions of y* and integrate along the *y-axis*
how to find intersection between two curves
set the two functions equal to each other and solve for x
if cross sections are perpendicular to the y-axis
solve equation for x (integrals use dy)
if cross sections are perpendicular to the x-axis...
solve equation for y (integrals use dx)
if you have a vertical axis of revolution (revolving around the y-axis)...
solve for x in terms of y (dy)
if you have a horizontal axis of revolution (revolving around the x-axis)...
solve for y in terms of x (dx)
what should you be aware of when finding total distance?
solve the function for zeros. if there are any zeros, you must incorporate that into the intervals. this is because the particle changes directions at zeros (FIND WHERE PARTICLE CHANGES DIRECTIONS TOO!) (for example, if the zeros are t = 5 and t = 2, and the full interval is [1, 7], you must find total distance by splitting up the integrals -> 1 to 2 + 2 to 5 + 5 to 7)
average value of a function
the value of f(c) given in the MVT for integrals is called the *average value of f on the interval [a, b]* this can also be written as area/width
how to solve the following problem: a ball is thrown up from a window, 48 ft above ground level and reaches its max height after 3 seconds. how long will it take the ball to reach ground level?
we know that a(t) = -32, so integrate to find the velocity function. we are given that the max height is reached after 3 seconds, so plug 3 in for t into the velocity function to find C1 (should be 96). v(t) is -32t + 96. integrate to find position function p(t) and plug in 48 for C2 (our initial condition/height). solve for t when p(t) = 0.
how to use trapezoidal rule when x/t values in table do not increase consistently by the same number
you must use trapezoidal area formula for each individual x/t value -> h/2[1st base + 2nd base]
mean value theorem for integrals
If f is continuous on [a,b], then there exists a c in [a,b] such that integral from a to b of f(x) dx = f(c)(b-a).
volume by washer method
R = outer radius r = inner radius
net change theorem
The definite integral of the rate of change of a quantity F'(x) gives the total change, or net change, in that quantity on the interval [a,b].
