AP Calculus Unit 6 Integration and Accumulation of Change

University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2022
Views:

470

Pages:

3
Author:

accionox1v

Unit 6: Integration and Accumulation of Change ● Riemann Sums: area under the curves ○ Right Hand Sum ■ 1) Subtract the interval from each other and divide it by how many sub intervals that the question asks for. ■ 2) Start from the right most f(x) value and multiply it to the width. ■ 3) Repeat step 2 going from right to left ■ 4) Add the sum of all boxes together ○ Left Hand Sum ■ Do the same steps as the Right Hand sum but start from the left side and go from left to right ○ Trapezoidal Sum ■ 1) Subtract the interval from each other and divide it by the amount of sub intervals wanted ■ 2) Start from the left ■ 3) Width * (f(a)+f(b)) * ½ ■ 4) Repeat ■ 5) Add the sum of all trapezoids together ○ Midpoint Sum ■ The same as the left and right-hand sum, but the height of each rectangle will be the midpoint of the base. ● Integrals ○ Definite Integral: the area under a specific closed interval ○ Indefinite Integral: represents a family of solutions ■ The answer has “ + C” after it, or “plus a constant” ● Properties of Integration ● Fundamental Theorem of Calculus: if the function is continuous on the interval and f(x) is the antiderivative of f(x) then: ● Mean Value Theorem for Integrals: If the function is continuous on the integral then: ● Second Fundamental Theorem of Calculus: if the function is continuous on an open interval then: ● Transcendental Functions: ● U-Substitution: reverses the chain rule ■ 1) Set a part of the function equal to U and differentiate it to find du. ■ 2) Multiply by dx so that du will be isolated ■ 3) Substitute du and u into the function ■ 4) Take the integral of the function ■ 5) Substitute the equations of du and u back in. ● Double U-Sub: Same as regular u substitution, except now we will need to solve for x ■ 1) Use the u function that we found and solve it so that we have a function of x ■ 2) Substitute the du, u, and x into the function ■ 3) Take the integral ■ 4) Plug the equations back into the function ● Integrals of ex and eu