Chapter 6, Problem 6RE

Let $R$ be the region in the first quadrant enclosed by $y=x^2,y=2+x,$ and $x=0$ . In each part, set up, but do not evaluate, an integral or a sum of integrals that will solve the problem.

(a) Find the area of $R$ by integrating with respect to $x$ .

(b) Find the area of $R$ by integrating with respect to $y$ .

(c) Find the volume of the solid generated by revolving $R$ about the $x\text{-axis}$ by integrating with respect to $x$ .

(d) Find the volume of the solid generated by revolving $R$ about the $x\text{-axis}$ by integrating with respect to $y$ .

(e) Find the volume of the solid generated by revolving $R$ about the $y\text{-axis}$ by integrating with respect to $x$ .

(f) Find the volume of the solid generated by revolving $R$ about the $y\text{-axis}$ by integrating with respect to $y$ .

(g) Find the volume of the solid generated by revolving $R$ about the line $y=-3$ by integrating with respect to $x$ .

(h) Find the volume of the solid generated by revolving $R$ about the line $x=5$ by integrating with respect to $x$ .