7.1

Area of a RegionBetween Two Curves

= -

Area of region between f and g

= Area of regionunder f(x)

- Area of regionunder g(x)

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f (x) − g(x)[ ]a

b

∫ dx = ∫ −b

a

dxxf )( ∫b

a

dxxg )(

f

g

f

g

f

g

Ex. Find the area of the region bounded by the graphs of f(x) = x2 + 2, g(x) = -x, x = 0, and x = 1 .

f(x) = x2 + 2

g(x) = -x

Area = Top curve – bottom curve

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A = f (x) − g(x)[ ]a

b

∫ dx =

€

(x 2 + 2) − (−x)[ ]0

1

∫ dx =

=⎥⎦

⎤⎢⎣

⎡++

1

0

23

223

xxx

6

172

2

1

3

1=++

Find the area of the region bounded by the graphs off(x) = 2 – x2 and g(x) = x

First, set f(x) = g(x) tofind their points of intersection.

2 – x2 = x

0 = x2 + x - 2

0 = (x + 2)(x – 1)

x = -2 and x = 1

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2 − x 2( ) − x[ ]

−2

1

∫ dx= fnInt(2 – x2 – x, x, -2, 1) 2

9=

Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x

Again, set f(x) = g(x)to find their points of intersection.

3x3 – x2 – 10x = -x2 + 2x

3x3 – 12x = 0

3x(x2 – 4) = 0

x = 0 , -2 , 2

Note that the two graphs switch at the origin.

Now, set up the two integrals and solve.

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(3x 3 −12x)dx + (−30

2

∫−2

0

∫ x 3 +12x)dx =

24=€

f (x) − g(x)[ ]−2

0

∫ dx + g(x) −f (x)[ ]0

2

∫ dx

dxcurvebottomcurvetopAx

x

])()[(2

1

−=∫

dyurvectflecurverightAy

y

])()[(2

1

−=∫

Find the area of the region bounded by the graphs of x = 3 – y2 and y = x - 1

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A = (3 − y 2) − (y +1)[ ]−2

1

∫ dy

2

9=

Area = Right - Left