712
Chapter 6 | Applications of Integration
Solution The region is depicted in the following figure.
Figure 6.70 Finding the centroid of a region between two curves.
The graphs of the functions intersect at (−2, −3) and (1, 0), so we integrate from −2 to 1. Once again, for the sake of convenience, assume ρ =1. First, we need to calculate the total mass: m = ρ ∫ ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx
a b
1 ⎡
−2 1
⎣ 1− x 2 −( x −1) ⎤
= ∫
⎦ dx = ∫
(2− x 2 − x ) dx
−2
⎦ | −2 1
⎡ ⎣ 2 x − 1 3
x 2 ⎤
⎡ ⎣ 2− 1 3 −
⎤ ⎦ −
⎡ ⎣ −4+ 8 3 −2 ⎤
1 2
x 3 − 1 2
⎦ = 9 2 .
=
=
Next, we compute the moments: M x = ρ ∫ a b
⎛ ⎝ ⎡
⎞ ⎠ dx
1 2
⎤ ⎦ 2 −
⎤ ⎦ 2
⎡ ⎣ g ( x )
⎣ f ( x )
1 ⎛
⎞ ⎠ dx = 1
−2 1 ⎛
2
⎝ ⎛
⎞ ⎠
⎝ x 4 −3 x 2 +2 x ⎞
= 1 2 ∫
2 ∫
⎝ 1− x 2
−( x −1) 2
⎠ dx
−2
⎤ ⎦ | −2 1
⎡ ⎣ x 5 5
− x 3 + x 2
= 1 2
= − 27 10
and
b
M y = ρ ∫
x ⎡ ⎣ f ( x )− g ( x ) ⎤
⎦ dx
a
−2 1
−2 1
⎡ ⎣ (1− x 2 )−( x −1) ⎤
⎡ ⎣ 2− x 2 − x ⎤
= ∫
⎦ dx = ∫
x
x
⎦ dx
−2 1 ⎛
⎝ 2 x − x 4 − x 2 ⎞
= ∫
⎠ dx
⎤ ⎦ | −2 1
⎡ ⎣ x 2 − x
5
3
− x
=
= − 9 4 .
3
5
Therefore, we have
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