Chapter 6.3, Problem 1QCE

Let $R$ be the region between the $x\text{-axis}$ and the curve $y=1+\sqrt{x}$ for $1\le x\le 4$ .

(a) For $x$ between $1$ and $4$ , the area of the cylindrical surface generated by revolving the vertical cross section of $R$ at $x$ about the $y\text{-axis}$ is $\text{\_\_\_\_\_\_\_}$ .

(b) Using cylindrical shells, an integral expression for the volume of the solid generated by revolving $R$ about the $y\text{-axis}$ is $\text{\_\_\_\_\_\_\_}$ .