U6: Integration and Accumulation of Change

Riemann Sums

You use Riemann sums to find the actual area underneath the graph of f(x).

Example: f(x) = x2 + 1 on [0, 2] with 4 equal subintervals

Left Riemann Sum

Left sum is an underestimate.

Right Riemann Sum

Right sum is an overestimate. 

Midpoint Sum

Trapezoidal Sum

Fundamental Theorem of Calculus (Definite Integration)

The area under the curve of derivatives of F from A to B is equal to the change in y-values of the function F from A to B, given f is:

● Continuous in interval [a, b]

● F is any function that satisfies F(x) = f(x)

What is an indefinite integration?

Given y' or f'(x), the anti-derivative can be thought of as the original function, f(x). Integration

is used to find the original function.

● The operation of finding all solutions to this equation is called antidifferentiation or indefinite integration.

● Detonated by an integral sign: ∫

v = ∫ f '(x) dx = f(x) + c

● f '(x) = integrand

● dx = variable of integration

● f(x) = antiderivative

● c = constant of integration

● ∫ = integral

Reminder: ALWAYS add + C when you’re solving for an INDEFINITE integral!

Reminder: Differentiation and integration are inverses!

Basic Integration Rules (w/ examples)

*K=constant/number

Antiderivative Trig Function

Examples with trig functions

∫ 2sinx dx = 2 ∫ sinx dx = 2(-cosx + c) = -2cosx + c

C is still a constant when multiplied by 2 

● HINT: How I memorize antiderivatives by using derivatives of trigonometric functions.

○ EX: d/dx sinx = cosx and for the antiderivative, you just switch the two trigonometric functions and add +C since it’s an indefinite integration.

○ EX: d/dx cscx = -cscxcotx and for the antiderivative, just switch the two trigonometric functions and add +c since it’s an indefinite integration. Also, if the derivative was negative, then the anti-derivative is also negative!

Integration by U-substitution

Reminder: Don’t forget to go back and replace u and add +C

Natural Log Function for Integration (Log rule for integration)

Integrals of the 6 Basic Trig Functions

● HINT: For ∫ tan u du, memorize like this: ∫ tan u du = ∫ sin u/cos u du because of the trigonometric identities. After that, just do u-substitution with cos u being u.

● If you work it out, it looks like this:

Integration rule for “e”

● With e, it’s just the same thing as regular u-substitution but with the additional ‘e’.

Integration Rule for Exponential Functions

Example: ∫7-xdx → - ∫7udu → -1/ln(7) * 7u + C → -7-x/ln7 + C

● u = -x → du = -dx → -du = dx