$($b$)5$ pointsl n$.$

Question $1q,25$ points

$($a$)[8$ points$]$ Find the highest and lowest values, and the points at which these occur for the function $f(x,y,z)=xy{z}^2$ on the sphere $x^2{y}^2{z}^2=1.$

$($b$)[9$ points$]$ Consider the double integral

${\displaystyle \underset{0}{\overset{2\sqrt[\phantom{2}]{ln3}}{}}}{\displaystyle \underset{\frac{0}{2}}{\overset{\sqrt[\phantom{2}]{ln3}}{}}}{e}^{x^2}dxdy$

i$.$ Sketch the region of integration.

ii$.$ Reverse the order of integration.

iii. Evaluate the integral that you found.

$($c$)$ points

$($b$)[5$ points$]$ Find the integral that calculates the volume enclosed between the plane

$x+2y+4z=8$

and the coordinate axes in the first octant. $($Do not evaluate the integral.$)$

$($c$)[15$ points$]$ Use an appropriate change of variables to evaluate the integral

${}_R(x-y{)}^{2}dA$

where the region $R$ is the closed quadrilateral with vertices $(0,0),(1,1),(2,0),$ and $(1,-1).$