Lesson 15-3

Example 1
Use limits to find the area of the region between the graph of y = x3 and the x-axis from
x = 0 to x = 2. That is, find .

First find x.
x	= 
	=  or 

Then find xi.
xi	= a + ix
	= 0 + i   or 		

Now we can calculate the integral that gives the area.

	= 	f(xi) = xi3
	= 	xi = , xi = 
	= 	Multiply.
	= 		
	= (13 + 23 +  + n3)	Factor.
	=   	13 + 23 +  + n3 = 
	= 	Multiply.
	= 4 +  + 	Divide by n2.
	= 	Limit theorems from Chapter 12
	= 4 + (8)(0) + (4)(0) or 4		

The area of the region is 4 square units.

Example 2
Use limits to find the area of the region between the graph of y = x4 and the x-axis from 
x = 1 to x = 3.

First find the area under the graph from x = 0 to x = 3. Then subtract from it the area under the graph from x = 0 to x = 1. In other words,  =  - .

For , a = 0 and b = 3, so x =  and xi = .	

For , a = 0 and b = 1, so x =  and xi = .

The area of the region between the graphs of y = x4 and the x-axis from x = 1 to x = 3 is 
 -  or 48 square units.

Example 3
Refer to the application at the beginning of Lesson 15-3 in your book. Before this company grew to its current size, it manufactured only 500 pairs of shoes per day. How much have costs to produce this shoe increased since then if they currently produce 2000 shoes per day?

The cost is given by  where f(x) = 20 - 0.004x is the marginal cost function.

a = 500 and b = 2000, so x =  and xi = 500 + 

This increase in production has cost this company $22,500.