The Area Between Two Curves Lesson 6.4

2 When f(x) < 0 Consider taking the definite integral for the function shown below. The integral gives a negative area (!?)  We need to think of this in a different way a b f(x)

3 Another Problem What about the area between the curve and the x-axis for y = x 3 What do you get for the integral? Since this makes no sense – we need another way to look at it

4 Solution We can use one of the properties of integrals We will integrate separately for -2 < x < 0 and 0 < x < -2 We take the absolute value for the interval which would give us a negative area.

5 General Solution When determining the area between a function and the x-axis  Graph the function first  Note the zeros of the function  Split the function into portions where f(x) > 0 and f(x) < 0  Where f(x) < 0, take absolute value of the definite integral

6 Try This! Find the area between the function h(x)=x 2 – x – 6 and the x-axis  Note that we are not given the limits of integration  We must determine zeros to find limits  Also must take absolute value of the integral since specified interval has f(x) < 0

7 Area Between Two Curves Consider the region between f(x) = x 2 – 4 and g(x) = 8 – 2x 2 Must graph to determine limits Now consider function inside integral  Height of a slice is g(x) – f(x)  So the integral is

8 The Area of a Shark Fin Consider the region enclosed by Again, we must split the region into two parts  0 < x < 1 and 1 < x < 9

9 Slicing the Shark the Other Way We could make these graphs as functions of y Now each slice is  y by (k(y) – j(y))

10 Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral is Known as an improper integral To solve we write as a limit (if the limit exists)

11 Improper Integrals Evaluating Take the integral Evaluate the integral using b Apply the limit

12 Assignments Lesson 6.4 Page 243 Exercises 1, 5, 9, … 49 (every other odd)