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Alenkasestr [34]
3 years ago
14

Evaluate the triple integral_E x^6e^y dV where E is bounded by the parabolic cylinder z = 25 - y^2 and the planes z = 0, x = 5,

and x = -5.
Mathematics
1 answer:
Nimfa-mama [501]3 years ago
4 0

Nothing crazy here, just a matter of figuring out the limits of integration.

\displaystyle\iiint_Ex^6e^y\,\mathrm dV=\int_{-5}^5\int_{-5}^5\int_0^{25-y^2}x^6e^y\,\mathrm dz\,\mathrm dy\,\mathrm dx

=\displaystyle\int_{-5}^{-5}\int_{-5}^5x^6e^y(25-y^2)\,\mathrm dy\,\mathrm dx

=\displaystyle\left(\int_{-5}^5x^6\,\mathrm dx\right)\left(\int_{-5}^5e^y(25-y^2)\,\mathrm dy\right)=\boxed{\frac{625,000(3+2e^{10})}{7e^5}}

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