Unit 6
integral sin(x) dx
-cos(x) + C
integral csc^2(x) dx
-cot(x) + C
integral csc(x) cot(x) dx
-csc(x) + C
general solution for inscribed or circumscribed rectangles
1. Draw a rough graph of the function over the interval. 2. Use formula ∆x = b−a/n to determine each subinterval length. 3. Compute x-coordinates of rectangles at either left-end or right-end. 4. Compute the areas of each rectangle (inscribed or circumscribed). 5. Find the summation of the approximated areas of the rectangles.
procedure for u-substitution for indefinite integrals
1. Look for a piece of the function whose derivative is also in the function. If you're not sure what to use, try the denominator or something being raised to a power in the function. 2. Set u equal to that piece of the function and take the derivative with respect to the variable (this will become du) 3. Use your u and du expressions to replace parts of the original integral, and your new integral will be much easier to solve. It may look like this, for example: ∫ u² du 4. In some cases, the du will not be the actual derivative that is found in the original function given in the problem. In this case, you may have to balance out the equation (that is, making du actually equal whatever derivative is found in the original function). For example, if du = 2x dx, but the derivative in the function is only x, you would need to divide by 2 to both sides and end up getting 1/2du = x dx.
tricks for integration
1. factor out/distribute 2. separate fractions (for example, there may be an x² in the denominator) 3. rewrite according to rules
writing a riemann sum in summation notation
1. find width of each rectangle on [a, b] → ∆x = b - a/n 2. find the right endpoint of each rectangle → xκ = a + ∆x × κ 3. plug in xκ into f(x). look at image to write the summation notation. 3. RHS: k values go from 1 to n 4: LHS: k values go from 0 to n - 1
area of trapezoid
A=½(h)(b₁+b₂) *h = length on the x-axis b₁ = y-value of first point b₂ = y-value of second point
integral 0 dx
C
what are two cases in which f(x) does NOT equal F'(x)? (the original derivative function does not equal the derivative of the antiderivative!)
CASE 1: lower bound is a constant, upper bound is a function of x CASE 2: both lower and upper bounds are a function of x
fundamental theorem of calculus part 2
F(b) - F(a) is the total/net change, or displacement in the antiderivative of f over the same interval. F(x) is the antiderivative
the substitution method for indefinite integral
Given: ∫ f(g(x)) (g'(x)) dx STEP 1: Let u = g(x) , then du = g′(x) STEP 2: Find the antiderivative of f STEP 3: Substitute g(x) back for u
definite integral definition
Let f be continuous on [a, b] and f(x) ≥ 0 for all x in the interval. Let [a, b] be partitioned into n subintervals of equal length ∆x = b−a/n. Then, the definite integral of f over a, b is given by:
indefinite integral
The family of all antiderivatives of a function f(x) is the indefinite integral of f with respect to x and is denoted by f(x) dx If F is any function such that F'(x) = f(x), then f(x) dx = F(x) + C, where C is an arbitrary constant (this is the general solution, as there are no specified limits).
substitution in definite integrals
When we integrate over an interval [a, b], the values of a and b are x-values. The use of u-substitution requires a change in [a, b] to [g(a), g(b)]. After identifying u and du, find the new upper and lower bounds by plugging each of them into the function that equals u. Then, rewrite the integral so that it takes ∫ u du as the base and evaluate using the specific limits
integral a^x dx
a^x / ln(a) + C
how to find general anti-derivative
add 1 to the exponent and then divide by that new power. add C at the end if there are no specified limits.
F(x)
anti-derivative
riemann sums give us an...
approximation of the area under the curve
area under the curve
area between the graph and the horizontal x-axis
definite integral notation
b = end point a = start point f(x) = integrand, or derivative function
when taking the integral of tan(x) or cot(x)...
change them to be in terms of sin(x) and cos(x), and then set u equal to the denomenator
The area under f, the derivative function, on some interval [a, b] accumulates at a decreasing rate so F, the antiderivative function, must be _______________________________ on that interval.
concave down
If the area under f, the derivative function, accumulates at an increasing rate, we know that the graph of F, the antiderivative function, must be ________________________________ on some interval.
concave up
f(x)
derivative
semi-circle
divide area of circle (πr²) by 2
integral e^x dx
e^x + C
odd function
f(-x) = - f(x) , symmetric with origin area would be *zero*
even function
f(-x) = f(x) , symmetric with y-axis add up both sides to get area
how to evaluate the integral of an absolute value function
find the zero(s) of the function by setting it equal to 0. then, create create two integrals that are added to each other, in which the lower bound of the first integral is the lower bound of the given problem and upper bound is the zero value; the lower bound of the second integral is the zero value again, and the upper bound is the upper bound of the given problem. evaluate.
fundamental theorem of calculus part 1
if f(x) is continuous on [a, b] then, g(x) = x∫a f(t)dt is continuous on [a, b] and it is differentiable on (a, b), also g'(x) = f(x). g is an antiderivative of f
when taking the integral of sec(x) or csc(x)...
in the case of sec(x) multiply by sec(x)tan(x)/sec(x)tan(x) OR in the case of csc(x) multiply by csc(x)+cot(x)/csc(x)+cot(x). then set u equal to the denomenator
If the area under f, the derivative function, is positive on some interval [a, b], we know that the graph of F, the antiderivative function, must be ____________________________________ on the interval [a, b].
increasing
what happens to the integral when b is less than a (end point is less than start point)?
it has to go backwards, meaning that the integral must turn into negative and b and a are flip flopped.
integral k f(x) dx
k ∫ f(x) dx
the definite integral of a constant, k, is just...
k(b - a)
integral k dx
kx + C
integral of du/u
ln|u| + C
integral 1/x dx
ln|x| + C
complete the square
means to rewrite as (x + ___)² + ____, where you must evaluate (b/2)²
regardless of concavity, RHS will always be...
overestimate for increasing functions underestimate for decreasing functions
left riemann sum (decreasing)
overestimates
right riemann sum (increasing)
overestimates
how do you find C, the constant?
plug in f(x) and x, then solve for C
integral sec(x)tan(x) dx
sec(x) + C
integral cos(x) dx
sin(x) + C
what do you do to the area when part of the graph is under the x-axis?
subtract the area underneath the x-axis from the area above the x-axis.
integral sec^2(x) dx
tan(x) + C
if the derivative function f has an extrema at some point...
the antiderivative g(x) = x∫0 f(t)dt has an inflection point
when the derivative function f changes signs (crosses the x-axis)...
the antiderivative g(x) = x∫0 f(t)dt is an extrema at that point
when the derivative function f is decreasing...
the antiderivative g(x) = x∫0 f(t)dt is concave down
when the derivative function f is increasing...
the antiderivative g(x) = x∫0 f(t)dt is concave up
when the derivative function f is negative...
the antiderivative g(x) = x∫0 f(t)dt is decreasing
when the derivative function f is positive...
the antiderivative g(x) = x∫0 f(t)dt is increasing
what should you not forget when using trapezoidal sums?
the area of a trapezoid; all points are used; the points inside (excluding the endpoints) must be multiplied by *2*
RHS uses all points except for...
the first point
LHS uses all points except for...
the last point
the definite integral can be used to determine...
the net area (displacement) bounded by the x-axis, the function f(x) and vertical lines x = a and x = b
the integral of the velocity function v(t) dt represents the...
the total change in the position over some interval
If r(t) represents the rate at which your pizza changes temperature in a 475°F pizza oven 5 minutes after the pizza had been placed into the oven to cook, the integral of r(t) dt on [5, 20] is...
the total change in the temperature of the pizza, in degrees Fahrenheit, from time t = 5 minutes to t = 20 minutes
regardless of concavity, LHS will always be...
underestimate for increasing functions overestimate for decreasing functions
left riemann sum (increasing)
underestimates
right riemann sum (decreasing)
underestimates
As k takes on different values, the functions graph as _________________________ transformations of one another, each containing the particular point ____________. Even though the lower bound changed is F′(x) = f(t) ? _______________________
vertical (k, 0) yes
to find left-hand or right-hand sums of *rectangles*...
width × [all y-points] *one y-point may not be included according to LHS or RHS estimations
integral [f(x) plus/minus g(x)]
∫ f(x) dx ± ∫ g(x) dx
