Q 43 - Calculus

Question

The iterated integrals use spherical coordinates. Describe the solids determined by the limits of integration. 

${\int }_{\frac{\pi }{2}}^{\pi }{\int }_0^{2\pi }{\int }_0^{2}f\left(\rho ,\theta ,\varphi \right){\rho }^2\sin \varphi d\rho d\theta d\varphi$

Step-by-Step Solution

Answer

The region represents sphere with second quadrant with the hemisphere below the $xy-$plane of the sphere with radius 2 centered at origin

Step 1:Given information

The given expression is ${\int }_{\frac{\pi }{2}}^{\pi }{\int }_0^{2\pi }{\int }_0^{2}f\left(\rho ,\theta ,\varphi \right){\rho }^2\sin \varphi d\rho d\theta d\varphi$

Step 2:Simplification

According to the order of integration

$\rho =0 to \rho =2$

width="141" style="max-width: none;" $\Rightarrow \sqrt{x^2+y^2+z^2}=2$

$\Rightarrow x^2+y^2+z^2=2^2$

This is a sphere equation

$\Rightarrow \theta$ varies from $\theta =0 to \theta =2\pi$

It means it is in the second quadrant

$\Rightarrow \varphi$ varies from $\frac{\mathrm{\pi }}{2} to \mathrm{\pi }$ these the region represents the hemisphere below the xy -plane of the sphere with radius 2 centered at the origin