Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows pages 1-3. to view the full**9 pages of the document.**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(x)g(x)f

the area highlighted in green that we want

○What to do?

○Take the area of (x) g(x)f−

Area Between Curves

Section 6.1

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

the curve of would give us the desired area.(x)f

●What would this give us?

○ (x) g(x)dxA = ∫

b

a

f−

○ OP (x) BOT (x)dxA = ∫

b

a

T−

■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 and 2x y= xy = 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

○x x2 = 2

○ x 2x0 = 2−

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○ x(x 2)0 = −

○, 2x= 0x=

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

●x x dx

∫

2

0

2 − 2

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

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

● x x dxA = ∫

2

0

2 − 2

○x(− x)= 23

13

○2 2 ) (0 0 )( 2−3

13− 2−3

13

○= 3

4

Example #2

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

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