$1.$Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in $x.$

$cos(si{n}^{-1}(\frac{7}{x}))$

$cos(si{n}^{-1}(\frac{7}{x}))=$

$($Type an exact answer, using radicals as needed.$)$

$2.$ Evaluate the integral ${\displaystyle \underset{0}{\overset{6\sqrt[\phantom{2}]{2}}{}}}\frac{7ds}{\sqrt[\phantom{2}]{144-s^2}}$

${\displaystyle \underset{0}{\overset{6\sqrt[\phantom{2}]{2}}{}}}\frac{7ds}{\sqrt[\phantom{2}]{144-s^2}}=$

$($Type an exact answer, using $$ as needed.$)$

$3.$ Evaluate the following derivative.

$\frac{d}{dx}(ln(7co{s}^{6}x))$

$\frac{d}{dx}(ln(7co{s}^{6}x))=$

$4.$Evaluate the integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(e^{8x}+9e^{-x})dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(e^{8x}+9e^{-x})dx=$

$5.$ Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the y $-$axis.

$y=13-x,y=0,x=1,$ and $x=3$

a$.$ Set up the integral that gives the volume of the solid using the shell method. Select the correct choice below and fill in the answer boxes to complete your choice. $($Type exact answers.$)$

b$.$ The volume is

$($Type an exact answer.$)$