1. The Area Between Two Curves
Lesson 6.4

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

3. 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)

4. 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

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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))

11. 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)

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

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