Let $f$ and $g$ be the functions defined by $f(x) = 1 + x + e^{x^2 - 2x}$ and $g(x) = x^4 - 6.5x^2 + 6x + 2$. Let $R$ and $S$ be the two regions enclosed by the graphs of $f$ and $g$ shown in the figure above. (a) Find the sum of the areas of regions $R$ and $S$. (b) Region $S$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. Find the volume of the solid. (c) Let $h$ be the vertical distance between the graphs of $f$ and $g$ in region $S$. Find the rate at which $h$ changes with respect to $x$ when $x = 1.8$.
Let $f$ and $g$ be the functions defined by $f(x) = 1 + x + e^{x^2 - 2x}$ and $g(x) = x^4 - 6.5x^2 + 6x + 2$. Let $R$ and $S$ be the two regions enclosed by the graphs of $f$ and $g$ shown in the figure above.
(a) Find the sum of the areas of regions $R$ and $S$.
(b) Region $S$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. Find the volume of the solid.
(c) Let $h$ be the vertical distance between the graphs of $f$ and $g$ in region $S$. Find the rate at which $h$ changes with respect to $x$ when $x = 1.8$.