Math $152$ Workshop $12$

Let $C$ be the curve parametrized by $d\frac{y}{d}xt=\frac{}{4}Cx=e^{2t}-2t,y=4e^t,1t3CCxc(t)=(tant,sect),-\frac{}{2}.$

$(a)$ Find $d\frac{y}{d}x.$

$(b)$ Find $an$ equation $of$ the tangent line when $t=\frac{}{4}$

Let $Cbe$ the curve parametrized $by$ the equations

$x=e^{2t}-2t,y=4e^t,1t3$

$(a)$ Find the arc length $ofC.$

$(b)$ Set $up(butdo$ not evaluate$)an$ integral for the surface area $of$ the surface obtained $by$ rotating $C$ about the $x-$axis.

$3.$ A partial solar eclipse looks like this, where dark circle $(the$ moon$)$ partially covers the white circle $(the$ sun$).$

Assume each circle has a radius $of1.(The$ moon $is$ much smaller than the sun, but $itis$ also much closer $tous,so$ they appear about the same size.$)$

Question: $In$ the eclipse shown $in$ the drawing above, what $is$ the area $of$ sun that remains exposed during the eclipse $(i.e.,$ the area that $is$ not covered $by$ the moon$)?$

Here's a hint: You can find the area $of$ a half circle without any calculus, $so$ the trick $is$ using integration $in$ polar coordinates $to$ find the area $of$ the striped regions.

$1$