Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows pages 1-2. to view the full**8 pages of the document.**Section 6.1 – Area between Curves

Find the Area of the Region

●What we want to do is take the integral, which would give us the area under the curve

●However, if we were to take the integral of () or () only, you wouldn’t quite get

the area highlighted in green that we want

○What to do?

○Take the area of () − ()

○Why? Because the area under the curve of () subtracted from the area under

the curve of () would give us the desired area

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●What would this give us?

○ = ∫

() − ()

○ = ∫

() − ()

■ In other words, the area between the curves is the area of the top curve

minus the bottom curve

Example #1

●Find the area between = 2 and = 2

●Firstly, we want to define the boundaries that we will take the area under

○Find the intersection points of the two functions

○Set the functions equal to each other to do so

○2 = 2

○0 = 2 − 2

○0 = ( − 2)

○ = 2, = 0

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○So we will take the integral from [0, 2]

●∫2

02 − 2

○We know that we can compile the new integral like so because from the graph,

we can clearly see that = 2 is clearly on the top

● = ∫2

02 − 2

○= 2(−1

33)

○(22−1

323)− (02−1

303)

○= 4

3

Example #2

●Find the area between the curves = 4, = 2, = 1

●Define the boundaries

○2 = 4

○ = 2

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