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h𝑊 𝑎𝑡 𝑖𝑠𝑔′ (𝑒 )?

8.2B – Area between to curves

•Goal: find the area by using horizontal strips rather than vertical.

y x

2y x

If we use the method we learned in 8.2A, we would have to integrate in two parts:

dx

dx

2 4

0 2 2x dx x x dx

Example

y x

2y x dy

We can find the same area using a horizontal strip.

Notice, now the horizontal strips will touch both curves no matter where we draw them, so we don’t need two integrals.

But there is an easier way!

y x 2y x

2y x 2y x

Since 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 dy

y x

2y x

dy

2 2

02 y y dy

length of strip width of strip

22 3

0

1 12

2 3y y y 8

2 43

10

3

Solution

Limits are y values.

Let’s do one together!

• Find the area of the region enclosed by

•Woa! We have Let’s solve it for y.

•No graph all three curves.

Solution

• Easier to use dy so you don’t have to split it up into 2 integrals.

• So: Solve each equation for x.

• (Already given)

Continued…

•Bounds: We can see the lower bound is y=-1, and the upper bound we have to find the intersection. y=1.793003715

•Use calculator: Area is about 4.21 units squared.

Homework