AP Calculus Unit 8 Application of Integration

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

2022
Views:

37

Pages:

2
Author:

trescandogss

Unit 8: Application of Integration ● Particle Motion o To determine total distance traveled: ■ 1) When given the position function, take the derivative ■ 2) Find where the derivative function equals zero ● This is where the particle changes direction ■ 3) Create a Wiggle chart ● If the test point results in a positive number, the particle is moving to the right; if negative, then left ■ 4) Plug in two endpoints and the points where f’(x) is zero to the original position equation ● This will give you the distance the particle moved in each direction ■ 5) Add together all the distances ○ If asking to find the displacement, make sure you consider the directions of the particles ■ If it is moving in the negative direction, subtract that distance. ● Average Value over a closed interval o Average Value Theorem: If the function is integrable on the interval then: o b and a are the endpoints of the closed interval ● Volume by cross-sectional area - Using integrals to find volume ○ Disc Method: function is rotated around x axis to form a solid ■ In order to find the area of a circle, you would use π*radius^2 ● But for this function, the height (radius) would be the function f(x), so, A=π* f(x)2 dx ● This would only give us the area at one specific x value, so, we must take the integral of that equation in order to find the volume of the whole shape → iπ * f(x)^2 dx ○ Washer Method: volume between two functions rotated around the x axis ■ π * (g(x)^2 - f(x)^2) dx ● g(x) is the function that is greater than f(x) ● Subtracting the two will give you the volume of just the area between the two functions.