U8: Applications of Integration

Average Value

● To find the average value, integrate the function by using the fundamental theorem of calculus

● After that, divide the answer by the length of the interval

Total Displacement

● The difference between the starting position and ending position

● Interval [a, b]

● Can be negative

● Formula: 

Integral A B v(t)dt = s(b) - s(a)

Example: What is the object’s displacement on the closed interval [0,2]

● s(t)=2t3 - 12t + 6

● s(b) - s(a) → s(2) - s(0) = [2(2)3 - 12(2) + 6] - [2(0)3 - 12(0) + 6]

● -2 - 6 = -8

Total Distance

● Total distance traveled by a particle is the sum of the amounts it displaces between the start, all of the stop(s), and the end.

● Distance can’t be negative

● Formula: ∫ba |v(t)|dt

● Example:

Area of a Region Between Two Curves

Area of region between f and g = area of region under f(x) - area of region under g(x)

∫ B A [f(x) - g(x)]dx = ∫ B A f(x)dx - ∫ B A g(x)dx

The Disk Method

If a region in the plane revolves about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest solid is a right circular cylinder or disk, which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle.

The horizontal axis of revolution

● Rotate Around x-axis

The vertical axis of revolution

● Rotate Around y-axis

The Washer Method

Horizontal Line of Rotation

π∫ba[(furthest equation) - (line of rotation)]2 + [(closest equation) - (line of rotation)]2dx

Vertical Line of Rotation

π∫dc[(furthest equation) - (line of rotation)]2 + [(closest equation) - (line of rotation)]2dy

The Washer Method Calculating Volume Using Integration

Step One: Draw a picture of your graph→ shade appropriate region

Step Two: Identify whether you are rotating about a vertical or horizontal line

● Vertical

○ Get everything in terms of y

● Horizontal

○ Get everything in terms of x

Step Three: Set up your integral

π∫ba[(furthest equation) - (line of rotation)]2 + [(closest equation) - (line of rotation)]2dx

Step Four: Simplify

Step Five: Integrate Definite Integral