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OLEGan [10]
3 years ago
15

Evaluate the triple integral. 5x dv, where e is bounded by the paraboloid x = 7y2 + 7z2 and the plane x = 7. e

Mathematics
1 answer:
Mrac [35]3 years ago
4 0

Convert to cylindrical coordinates:

x=x

y=r\cos\theta

z=r\sin\theta

Then E is the set of points (r,\theta,x) such that 0\le r\le1, 0\le\theta\le2\pi, and 7y^2+7z^2\le x\le7 or 7r^2\le x\le7.

Now

\displaystyle\iiint_E5x\,\mathrm dV=5\int_0^{2\pi}\int_0^1\int_{7r^2}^7xr\,\mathrm dx\,\mathrm dr\,\mathrm d\theta=\boxed{\frac{245\pi}3}

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