Area Between Two CurvesHow do you find the area between two curves?

What is the area underneath the curve of from to ?

∫0

4

(5𝑥−𝑥2)𝑑𝑥 ¿ [ 52 𝑥2− 13 𝑥3]04

¿18 23

What is the area underneath the curve of

from to ?∫0

4

𝑥 𝑑𝑥 ¿ [ 12 𝑥2]04

¿8

What is the area between the curves of and from to ?

∫0

4

(5𝑥−𝑥2)𝑑𝑥−∫0

4

𝑥 𝑑𝑥¿18 23−10=823∫

0

4

(5𝑥−𝑥2−𝑥 )𝑑𝑥∫0

4

(4 𝑥−𝑥2)𝑑𝑥

−

− ¿¿

Area between two curves:

When

𝑨=∫𝒂

𝒃

[ 𝒇 (𝒙 )−𝒈 (𝒙 ) ]𝒅𝒙

Example 1Find area of the region bounded by the graphs of and between and .

¿∫0

1

(𝑥2+𝑥+2 )𝑑𝑥

Find area between and .

Since between

Find where the graphs intersect.

Example 2: Region between two intersecting graphs.

− (𝑥2+𝑥−2 )=0

If curves intersect at more than one point.

Horizontal Area• Sometimes it will be more convenient to find

the area of a region by differentiating with respect to y instead of x.

-Setup your integral from bottom to top and use the y values instead of the x values.

-You will integrate (right function – left function).

∫−2

1

❑

Integrate this same function using .You must divide the integral into two parts.

Summary

1. Non-Intersecting Graphs

2. Intersecting Graphs

3. Graphs with Multiple Intersections

4. Finding Area Horizontally

Homework

Area Between Curves Section 7.1

(1-23 odd, 27, 29) (omit 11)

*For 13-21 odd, set up on paper but you may use your calculator to integrate.*

DELETED SLIDES

Example 2: Region between two intersecting graphs.

Find area between and .

Find the points where graphs intersect and integrate from the x

value of the left intersection to the x

value of the right intersection.