Quection $3(25$ marks$)$

a$)$ Given that ${\displaystyle \underset{-a}{\overset{\sqrt[\phantom{2}]{\sqrt[\phantom{2}]{4}{?}^{-2}}}{}}}{\displaystyle \underset{0}{\overset{2}{}}}(x^2+y^2{)}^{\frac{4}{2}}dxdy$

i$)$ Sketch the region $R.$

$[2$ marks$]$

ii$)$ Evalnate the integral by converting it to polar coordinates.

$[6$ marks$]$

b$)$ Given ${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{\sqrt[\phantom{2}]{5}}{\overset{1}{}}}\frac{y{e}^{x^2}}{x^3}dxdy.$ Calculate the iterated integral by first reversing the order of integration.

$[8$ marks$]$

c$)$ Given ${\displaystyle \underset{-3}{\overset{3}{}}}{\displaystyle \underset{-\frac{\sqrt[\phantom{2}]{9-x^2}}{\sqrt[\phantom{2}]{9-x^2}}}{\overset{\sqrt[\phantom{2}]{6x^2-y^2}}{}}}=\sqrt[\phantom{2}]{x^2+y^2+z^2}dzdy$ dx$.$ Evaluate the integral by using the method of spherical coordinates. $[$Hint: $x=sincos,y=sinsin,z=cos]$

$[9$ marks$]$

Question $4(25$ marks$)$

a$)$ Use Stoke's theorem to evaluate ${}_{}(grad\frac{}{b}ar(F))n^{}dS$ for the given vector field vec$(F)(x,y,z)=y{j}^{}-$hat$(j)+$xhat$(k),$ where $$ is the upper half surface of the sphere $x^2+y^2+z^2=1.$

$[8$ marks$]$

b$)$ By using Green's Theorem, evaluate $o{\displaystyle \underset{c}{\overset{\phantom{}}{}}}(x{e}^{-2x})dx+(x^4+2x^2{y}^2)dy$ where $C$ is the boundary of the region between the circles $x^2+y^2=1$ and $x^2+y^2=4.$

$($Hint: Use polar coordinates for the integration$)$

$[11$ marks$]$

c$)$ Given vec$(F)(x,y,z)=y^2{z}^3$hat$(i)+2x{z}^3{j}^{}+3x{y}^2{z}^2$hat$(k),$

i$)$ Show that $)=($ is a conservative vector field.

$[3$ marks$]$

ii$)$ If the potential function for vec$(F)$ is $f(x,y,z)=x{y}^2{z}^3+K,$ evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}Fdr$ where $C$ is the simple curve from $A(0,2,0)$ to $B(1,1,1).$

$[3$ marks$]$