On-line Math 21

On-line Math 21

6.1  Area between curves

Example 1 Find the areas enclosed by the curves y = x2 and y = 2x+1 .

Solution

The region is like this:

By ``the area enclosed by'' or ``the region bounded by'' we would mean the region trapped between those two curves, that name-brand ``swoosh'' region. To find the intersection points, which in this case define the first and last x you need for this region, you have to find where the curves have the same y for the same x , where
x2 = 2x+1,
which by the quadratic formula is at
x = 1±Ö2,
where x2-2x-1 = 0 .

Then, note that the top curve is the curve y = 2x+1 , and the bottom is y = x2 , in the region bounded by these two curves. By the formula, then,
Area
=
ó
õ 		1+Ö2

1-Ö2 
 ( (2x+1)-x2) dx
=
x2+x- x3
3
 ê
ê
ê 		1+Ö2

1-Ö2 
=
( 1+Ö2) 2+( 1+Ö2) - ( 1+Ö2) 3
3
- æ
ç
è 		( 1-Ö2) 2+( 1-Ö2) - ( 1-Ö2) 3
3
 ö
÷
ø
=
8
3
 Ö2
@
3.77

Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 3 Jan 2001, 23:40.