Evaluate

$$ B

e$($x$2+$ y$2+$ z$2)3/2$

where B is the ball:

B $=$

$($x$,$ y$,$ z$)$

$$x$2+$ y$2+$ z$24$

$.$

SOLUTION Since the boundary of B is a sphere, we use spherical coordinates:

B $=$

$(,,$ $)$

$02,02,0$ $$

$.$

In addition, spherical coordinates are appropriate because

x$2+$ y$2+$ z$2=2.$

Thus this equation gives

$$ B

e$($x$2+$ y$2+$ z$2)3/2$

$=$

$0$

$20$

$20$

e$(2)3/2$

$$d$$ d$$ d

$=$

$0$

$$d

$20$

d$$

$20$

$2$e$3$ d$$

$=$

$$cos$($$)$

$0$

$(2)$

$20$

$=$

$.$