714
Chapter 6 | Applications of Integration
a b
m = ρ ∫
f ( x ) dx
2 ⎛
⎞ ⎠ dx
= ∫
⎝ 4− x 2
−2
⎤ ⎦ | −2 2
⎡ ⎣ 4 x − x
3
=
= 32 3 .
3
Next, we calculate the moments. We only need M x : M x = ρ ∫ a b ⎡ ⎣ f ( x ) ⎤ ⎦ 2 2 dx = 1 2 ∫ −2 2 ⎡ ⎣ 4− x 2 ⎤ ⎦ 2
−2 2 ⎛
⎝ 16−8 x 2 + x 4 ⎞
2 ∫
dx = 1
⎠ dx
⎦ | −2 2
⎡ ⎣ x 5 5
x ⎤
3 3 +16
− 8 x
= 1 2
= 256 15
.
Then we have
M x m = 256 15
y – =
8 5
· 3 32 =
.
The centroid of the region is (0, 8/5).
6.33 Let R be the region bounded above by the graph of the function f ( x ) =1− x 2 and below by x -axis. Find the centroid of the region.
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