Find a normal vector to the surface $xy-z^2=y^{2}z-1$ at the point $(7,3,2).$ Then find the equation of the tangent plane at this point.

Let $R$ be the region in the $xy-$plane bound by the lines $x+y=2,y-x=2,$ and $y=5.$ Set up a single iterated double integral to compute ${}_{R}f(x,y)dA$ using the order of integration $dxdy$ or $dydx($whichever one is appropriate$).$ Then compute the double integral using the function $f(x,y)=x^2.$

Reverse the order of integration and evaluate the integral: ${\displaystyle \underset{0}{\overset{4}{}}}{\displaystyle \underset{\sqrt[\phantom{2}]{y}}{\overset{2}{}}}3e^{(x^3)}dxdy.$

Consider the region $R$ bound the line $y=-x,$ the $x-$axis, and curve $y=\sqrt[\phantom{2}]{25-x^2}.($This is a "wedge" contained in the second quadrant.$)$ Write $R$ in set$-$builder notation using polar coordinates:

Convert the following double integral from Cartesian to polar coordinates. Then evaluate the integral:

${\displaystyle \underset{-3}{\overset{3}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{9-x^2}}{}}}sin(x^2+y^2)dydx$

Consider the region $R$ in the $xy-$plane bound by the curves $y=x^2$ and $y=6x-8.$ Let $D$ be the solid lying above the region $R$ and below the plane $z=x.$

$($a$)$ Use the appropriate double integral to compute the volume of the solid $D.$

$($b$)$ Use the appropriate triple integral to compute the volume of the solid $D.$

$($c$)$ Find the average value of the function $f(x,y)=x$ over the region $R.$

Consider the solid below the paraboloid $z=16-x^2-y^2$ and above the $xy-$plane. A cylindrical hole is cut through this solid using the cylinder $x^2+y^2=4,$ resulting in a new solid $D.$ Set up a double integral in polar coordinates for computing the volume of the solid $D,$ then compute the volume.

Consider the solid $D$ bound by the sphere $x^2+y^2+z^2=20$ and the paraboloid $z=x^2+y^2$ in the first octant. Set up a triple integral in cylindrical coordinates to compute the volume of this solid.

Let $D$ be the top half of a ball of radius $3$ centered at the origin. Find the average distance of points in $D$ from the origin using the appropriate triple integral in spherical coordinates.

Hint: Set up a function $f(,,)$ which gives the distance of the point $(,,)$ to the origin. Then use the triple integral formula for the average of a function.