Let $f(x) = e^{2x}$. Let $R$ be the region in the first quadrant bounded by the graph of $f$, the coordinate axes, and the vertical line $x = k$, where $k > 0$. The region $R$ is shown in the figure above. (a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of $R$ in terms of $k$. (b) The region $R$ is rotated about the $x$-axis to form a solid. Find the volume, $V$, of the solid in terms of $k$. (c) The volume $V$, found in part (b), changes as $k$ changes. If $\frac{dk}{dt} = \frac{1}{3}$, determine $\frac{dV}{dt}$ when $k = \frac{1}{2}$.
Let $f(x) = e^{2x}$. Let $R$ be the region in the first quadrant bounded by the graph of $f$, the coordinate axes, and the vertical line $x = k$, where $k > 0$. The region $R$ is shown in the figure above.
(a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of $R$ in terms of $k$.
(b) The region $R$ is rotated about the $x$-axis to form a solid. Find the volume, $V$, of the solid in terms of $k$.
(c) The volume $V$, found in part (b), changes as $k$ changes. If $\frac{dk}{dt} = \frac{1}{3}$, determine $\frac{dV}{dt}$ when $k = \frac{1}{2}$.