$($a$)(10$ points$)$ Evaluate

${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{si{n}^{-1}x}{\overset{\frac{}{2}}{}}}{e}^{cosy}cosydydx$

$($b$)(10$ points$)$ Evaluate

${\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{x^2}{\overset{4}{}}}\frac{x^{3}y}{\sqrt[\phantom{2}]{x^4{y}^2}}dydx$

$(20$ points$)$ Let

$S=\{(x,y,z)in{R}^3$:$x^2{y}^2{z}^2=1,z\frac{1}{2}\}$

be the cap of the unit sphere lying above the plane $z=\frac{1}{2}.$ Let $n$ denote the outward unit normal to $S.$ Consider the vector field

$F(x,y,z)=(-y{e}^z,x{e}^z,cos(xy))$

Evaluate

${}_S(gradF)*ndS$

$(20$ points$)$ Consider

$A=([1,2],[0,4])$

For each nonnegative integer $n,$ let $a_n$ and $b_n$ be real numbers satisfying

$A^n(\frac{a_n}{b_n})=(\frac{0}{1})$

Determine the interval of convergence of the power series:

${\displaystyle \underset{n=0}{\overset{}{}}}{a}_n{x}^n$Let ${}_n={\displaystyle \underset{k=1}{\overset{n}{}}}\frac{1}{k}-{\displaystyle \underset{1}{\overset{n}{}}}\frac{1}{t}dt.$ Prove that $\underset{n}{lim}{}_n$ exists and finite.

$($a$)$ Evaluate

${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{si{n}^{-1}x}{\overset{\frac{}{2}}{}}}{e}^{cosy}dydx$

$($b$)$ Evaluate

${\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{x^2}{\overset{4}{}}}\frac{x^3}{\sqrt[\phantom{2}]{x^4{y}^2}}dydx$

Let $S=\{(x,y,z)$:$x^2{y}^2{z}^2=1,z0\}$ and $n$ be the outward unit normal vector to $S.$ Evaluate

${}_S(gradF)*ndS$

where

$F=(e^{z^3-2z^{2}3z}x,sin(x^{2}yz)y2,e^{z^2}sin(z^2)x^3{y}^3)$

Let $S$ be the boundary of the area $T=\{(x,y,z)$:$0z4-x^2-y^2\}$ and $n$ be the outward unit normal vector to $S.$ Find the flux of the vector field

$F=(xzsin(yz)x^3,cos(yz)x^5{z}^3,3z{y}^2-e^{x^5{y}^7})$

through $S.$