Find the volume of the solid enclosed by : $z=7−x^2, z=−2, y=−1, y=4$. My answer is 180, the answer in my book is totally different.
This is my multiple integral:
$\int_{-1}^{4} ( \int_{-3}^{3}\left| 7-x^2-(-2)\right|dx )dy$
Volume of the solid enclosed by curves.
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integration
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0$x2$ or $x^2$ ? – 2017-01-11
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0$x^2$ i'm sorry – 2017-01-11
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0My answer is 180 too. – 2017-01-11
1 Answers
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First, find the area enclosed by a parabolic cross section in the $xz$-plane enclosed by the curves $$z = f(x) = 7-x^2, \quad z = g(x) = -2:$$ this is simply $$B = \int_{x=-3}^3 f(x) - g(x) \, dx = \int_{x=-3}^3 9-x^2 \, dx = 36.$$ Thus the total volume is simply the area of the base $B$ times the height of the parabolic cylinder, which is the distance between $y = -1$ and $y = 4$, which is $h = 5$. Thus the total volume is $V = Bh = 5(36) =180$, as you found.