Final Exam $-$ Summer $2023$ Calculus and Analytical Geometry IV $($Math$-250)$

Q$\mathrm{.}5)($a$)$ Evalute

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${\displaystyle \underset{-a}{\overset{a}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{a^2-y^2}}{\overset{\sqrt[\phantom{2}]{a^2-y^2}}{}}}{e}^{-(x^2{y}^2)}dxdy$

$($b$)$ Use the triple integration in cylindrical coordinate to find the volume of a solid that is bounded above by the hemisphere $z=\sqrt[\phantom{2}]{81-x^2-y^2},$ below by the $xy-$plane and laterally by the cylinder $x^2{y}^2=9.$

$34$ marks

Q$\mathrm{.}6)($a$)$ Let $T$ be the transformation from the uv$-$plane to xy $-$plane defined by the equations $u=\frac{x}{y}$ and $v=xy.$ Find the Jacobian of the transformation.

$($b$)$ Using this transformation evaluate

${}_{R}x{y}^{3}dA$

region R in the first quadrant enclosed by $y=x,y=3x,xy=1$ and $xy=4.$

$25$ marks