Consider the iterated triple integral ${\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{3}}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{3-x^2}}{}}}{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{4-x^2-y^2}}{}}}zdzdydx$ and the fol$-$

lowing iterated triple integrals:

$($a$){\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{\frac{\sqrt[\phantom{2}]{3}}{3}}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{4-r^2}}{}}}rzdzdrd$

$($b$){\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{3}}{}}}{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{4-r^2}}{}}}rzdzdrd$

$($c$){\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{3}}{}}}{\displaystyle \underset{1}{\overset{2}{}}}rzdzdrd$

$($d$){\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{\frac{}{3}}{}}}{\displaystyle \underset{sec()}{\overset{2}{}}}cos()ddd$

$($e$){\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}{\displaystyle \underset{0}{\overset{\frac{}{6}}{}}}{\displaystyle \underset{0}{\overset{2}{}}}{}^{3}sin()cos()ddd$

$($f$){\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}{\displaystyle \underset{0}{\overset{\frac{}{3}}{}}}{\displaystyle \underset{sec()}{\overset{2}{}}}{}^{3}sin()cos()ddd$

$(1)$ Which of the integrals $(a)-(f)$ above is the result of correctly converting the integral at the top of the page from rectangular coordinates to cylindrical coordinates?

$(2)$ Which of the integrals $(a)-(f)$ above is the result of correctly converting the integral at the top of the page from rectangular coordinates to spherical coordinates?