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

What is the area underneath the curve of 𝑦=5𝑥− 𝑥 2 from 𝑥=0 to 𝑥=4? 0 4 5𝑥− 𝑥 2 𝑑𝑥 = 5 2 𝑥 2 − 1 3 𝑥 3 0 4 =18 2 3 What is the area underneath the curve of 𝑦=𝑥 from 𝑥=0 to 𝑥=4? 0 4 𝑥 𝑑𝑥 = 1 2 𝑥 2 0 4 =8 What is the area between the curves of 𝑦=5𝑥− 𝑥 2 and 𝑦=𝑥 from 𝑥=0 to 𝑥=4? 0 4 5𝑥− 𝑥 2 𝑑𝑥 0 4 5𝑥− 𝑥 2 −𝑥 𝑑𝑥 0 4 4𝑥− 𝑥 2 𝑑𝑥 − 0 4 𝑥 𝑑𝑥 =18 2 3 −10=8 2 3

Note: Although Pink does not show up well on computer, does show up on projected image. − = − =

Area between two curves: When 𝑓 𝑥 ≥𝑔 𝑥 for all x between 𝑎,𝑏 Use highlighter to highlight 𝑓(𝑥) graph. 𝑨= 𝒂 𝒃 𝒇 𝒙 −𝒈 𝒙 𝒅𝒙

Example 1 Find area of the region bounded by the graphs of 𝑦= 𝑥 2 +2 and 𝑦=−𝑥 between 𝑥=0 and 𝑥=1. = 0 1 𝑥 2 +𝑥+2 𝑑𝑥 Don’t use class-time to solve. Just have the students set up the integral.

Find area between 𝑓 𝑥 =2 − 𝑥 2 and 𝑔 𝑥 =𝑥. Example 2: Region between two intersecting graphs. Find area between 𝑓 𝑥 =2 − 𝑥 2 and 𝑔 𝑥 =𝑥. Find where the graphs intersect. − 𝑥 2 +𝑥−2 = 0 Since 𝒇 𝒙 ≥𝒈 𝒙 between −𝟐,𝟏 −𝟐 𝟏 𝟐− 𝒙 𝟐 −𝒙 𝒅𝒙

−𝟐 𝟏 𝟐− 𝒙 𝟐 −𝒙 𝒅𝒙

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.

-You will integrate (right function – left function). -Setup your integral from bottom to top and use the y values instead of the x values. −2 1 -You will integrate (right function – left function).

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

Summary Non-Intersecting Graphs Intersecting Graphs Graphs with Multiple Intersections 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 𝑓 𝑥 =2 − 𝑥 2 and 𝑔 𝑥 =𝑥. Find the points where graphs intersect and integrate from the x value of the left intersection to the x value of the right intersection.