7.2Areas in the Plane

Gateway Arch, St. Louis, Missouri Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

21 2y x

2y x

How can we find the area between these two curves?

We could split the area into several sections, use subtraction and figure it out, but there is an easier way.

21 2y x

2y x

Consider a very thin vertical strip.

The length of the strip is:

1 2y y or 22 x x

Since the width of the strip is a very small change in x, we could call it dx.

21 2y x

2y x

1y

2y

1 2y ydx

Since the strip is a long thin rectangle, the area of the strip is:

2length width 2 x x dx

If we add all the strips, we get:2 2

12 x x dx

21 2y x

2y x

2 2

12 x x dx

23 2

1

1 123 2

x x x

8 1 14 2 23 3 2

8 1 16 23 3 2

36 16 12 2 36

276

92

The formula for the area between curves is:

1 2Areab

af x f x dx

We will use this so much, that you won’t need to “memorize” the formula!

y x

2y x

y x

2y x

If we try vertical strips, we have to integrate in two parts:

dx

dx

2 4

0 2 2x dx x x dx

We can find the same area using a horizontal strip.

dySince the width of the strip is dy, we find the length of the strip by solving for x in terms of y.y x

2y x

2y x

2y x

y x

2y x

We can find the same area using a horizontal strip.

dySince the width of the strip is dy, we find the length of the strip by solving for x in terms of y.y x

2y x

2y x

2y x

2 2

02 y y dy

length of strip

width of strip

22 3

0

1 122 3y y y

82 43

103

General Strategy for Area Between Curves:

1

Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.)

Sketch the curves.

2

3 Write an expression for the area of the strip.(If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y.

4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.)

5 Integrate to find area.