The functions $f$ and $g$ are given by $f(x) = \sqrt{x}$ and $g(x) = 6 - x$. Let $R$ be the region bounded by the $x$-axis and the graphs of $f$ and $g$, as shown in the figure above. (a) Find the area of $R$. (b) The region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 2$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose base lies in $R$ and whose height is $2y$. Write, but do not evaluate, an integral expression that gives the volume of the solid. (c) There is a point $P$ on the graph of $f$ at which the line tangent to the graph of $f$ is perpendicular to the graph of $g$. Find the coordinates of point $P$.
The functions $f$ and $g$ are given by $f(x) = \sqrt{x}$ and $g(x) = 6 - x$. Let $R$ be the region bounded by the $x$-axis and the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) The region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 2$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose base lies in $R$ and whose height is $2y$. Write, but do not evaluate, an integral expression that gives the volume of the solid.
(c) There is a point $P$ on the graph of $f$ at which the line tangent to the graph of $f$ is perpendicular to the graph of $g$. Find the coordinates of point $P$.