91Ó°ÊÓ

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{2}{3}, \frac{\pi}{2}, \frac{\pi}{6}\right) $$

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{2}{3}, \frac{\pi}{2}, \frac{\pi}{6}\right) $$

Find the radius of gyration about the indicated axis of the lamina that has the given shape and density. $$ x+y=a, a>0, x=0, y=0 ; \rho(x, y)=k \text { (constant); } x \text { -axis } $$

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(5, \frac{5 \pi}{4}, \frac{2 \pi}{3}\right) $$

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(5, \frac{5 \pi}{4}, \frac{2 \pi}{3}\right) $$

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(8, \frac{\pi}{4}, \frac{3 \pi}{4}\right) $$

A lamina has the shape of the region bounded by the graph of the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\). If its density is \(\rho(x, y)=1\), find: (a) the moment of inertia about the \(x\) -axis of the lamina, (b) the moment of inertia about the \(y\) -axis of the lamina, (c) the radius of gyration about the \(x\) -axis [Hint: The area of the ellipse is \(\pi a b]\), and (d) the radius of gyration about the \(y\) -axis.

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(8, \frac{\pi}{4}, \frac{3 \pi}{4}\right) $$

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{1}{3}, \frac{5 \pi}{3}, \frac{\pi}{6}\right) $$

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{1}{3}, \frac{5 \pi}{3}, \frac{\pi}{6}\right) $$