Let $e_r=($:$\frac{x}{r},\frac{y}{r},\frac{z}{r}$:$)$ be a unit radial vector, where $r=\sqrt[\phantom{2}]{x^2+y^2+z^2}.$

$($a$)$ Calculate the integral of $F=e^{-r}{e}_r$ over the upper hemisphere of $x^2+y^2+z^2=121$ with the normal pointing outward.

$($Give your answer in exact form. Use symbolic notation and fractions where needed.$)$

Hint I: The outward unit normal vector at any point $(x,y,z)$ on the sphere is $e_r($Midterm $1).$

Hint $2$: Don't forget the Jacobean.

Hint $3$: Please watch the second part of this vide: hups:$/$mediahubikuedu$\text{'}$media$//$I$\_$ca$4$kde$4$z

$F*e_r=$

$($Give your answer in exact form.$)$

${\displaystyle \underset{S}{\overset{\phantom{}}{}}}F*dS=$

$$

$($b$)$ Calculate the integral of $E=11e^{-r}{e}_r$ over the octant $x0,y0,z0$ of the unit sphere centered at the origin.

$($Give your answer in exact form. Use symbolic notation and fractions where needed.$)$

Hint: There are a few differences between this part and the previous part; note the radius of the sphere, the octant$($s$),$ and the vector field.

$E*e_r=$

$($Give your answer in exact form. Not that the radius of the surface has changed.$)$

${}_{S}E*dS$