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.
