The Area Between Two Curves Lesson 6.4

What If … ? We want to find the area between f(x) and g(x) ? Any ideas?

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)

Another Problem What about the area between the curve and the x-axis for y = x3 What do you get for the integral? Since this makes no sense – we need another way to look at it

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.

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

Try This! Find the area between the function h(x)=x2 – 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

Area Between Two Curves Consider the region between f(x) = x2 – 4 and g(x) = 8 – 2x2 Must graph to determine limits Now consider function inside integral Height of a slice is g(x) – f(x) So the integral is

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

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))

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)

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

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