Evaluate the following iterated integral by converting

Ответ:

IntellTanito

31.01.2022

Looks like the integral is

$\displaystyle\int_{-8}^8\int_0^{\sqrt{64-x^2}}\sin(x^2+y^2)\,\mathrm dy\,\mathrm dx$

The integration region is the upper half of a circle in the x-y plane centered at the origin with radius 8. So, in polar coordinates, the integral is

$\displaystyle\int_0^{2\pi}\int_0^8r\sin(r^2)\,\mathrm dr\,\mathrm d\theta=2\pi\int_0^8r\sin(r^2)\,\mathrm dr$

$=\pi\displaystyle\int_0^{64}\sin s\,\mathrm ds$

where we substituted s=r^2 and $\mathrm ds=2r\,\mathrm dr$ , which gives

$=\boxed{\pi(1-\cos(64))}$

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Ответ:

jasperzhouzihe3018

14.04.2021

Mathematics

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Evaluate the following iterated integral by converting to polar coordinates. ∫8 −8∫0^(64−x^2)1/2 sin(x^2+y^2) dydx

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