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7.2

Areas in the Plane

Gateway Arch, St. Louis, Missouri

Find the area under this curve.

Remember? An infinite number of rectangles.

Find the area under this curve.

We could simply subtract the smaller area from

the larger.

Or we could try looking at it this way...

How can we find the area between these two curves?

Consider a very thin vertical strip.

The length of the strip is

or

Since the width of the strip is a very small

change in x, we could call it dx.

How can we find the area between these two curves?

Since the strip is a long thin rectangle, the

area of the strip is

height

base

If we add all the strips, we get

(No Transcript)

The formula for the area between curves is

We will use this so much, that you wont need to

memorize the formula!

How can we find the area between these two curves?

(No Transcript)

If we try vertical strips, we have to integrate

in two parts

We can find the same area using a horizontal

strip.

Since the width of the strip is dy, we find the

length of the strip by solving for x in terms of

y.

length of strip

width of strip

General Strategy for Area Between Curves

Sketch the curves.

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.)

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.

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