1. The Area Between Two Curves
Lesson 7.5

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 = x3
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)=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

7. 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

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. Application
Suppose a company's costs and savings are changing – the rate of change functions are
We wish to know two things …
How long will they realize savings?
What will be the total amount of savings for this period?

11. Equilibrium Price Consider intersection of demand and supply curves
Called the equilibrium price
D(q)
Price at which consumers will purchase same quantity of product manufacturers want to sell
Price
S(q)
Quantity

12. Consumers' Surplus
Total of differences between equilibrium price and higher prices (some are willing to pay) is consumers' surplus
D(q)
Price
(q0, p0)
S(q)
Quantity

13. Producers' Surplus
Total of differences between equilibrium price and lower prices (manufacturers are willing to receive) is producers' surplus
D(q)
Price
(q0, p0)
S(q)
Quantity

14. Assignment
Lesson 7.5
Page 409
Exercises 1 – 39 odd