4. (a) ( 8 points) Find the volume of the solid that is bounded above by the plane $z=4$ and bounded below by $z=\sqrt{x^{2}+y^{2}}$. (b) ( 6 points) Find the area of the region in the first quadrant that lies within the cardioid $r=1-\cos \theta$. (c) ( 3 points) of the following vector valued functions, which ones trace out the part of the parabola $y=x^{2}$ from $(-2,4)$ to $(2,4)$ as $t$ ranges over the given domain? Explain your answer. (i) $\overrightarrow{\mathrm{r}}_{1}(t)=\left\langle e^{t}, e^{2 t} ight angle$, $-\ln (2) \leq t \leq \ln (2)$ (ii) $\vec{r}_{2}(t)=\left\langle t^{2}, t^{4} ight angle$, $-\sqrt{2} \leq t \leq \sqrt{2}$ (iii) Sloverrightarrow{\mathbf{r}}-{3}(t)=\left\langle 2 \sin t, 4(\sin t)^{2} ight angle$, $-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$ (d) (3 points) Give an example of a function $w=f(x, y, z) $ whose level surfaces at height $c$ are spheres of radius $c^{2}$ for all $c \in \mathbb{R}, c>0$. CS. JG. 135