Mathematics

Problem

Let

R

be the region in the

xy

-plane bounded by

y=\sqrt{x}

, the

x

-axis, and the vertical line

x=4

. The region

R

is revolved about the vertical line

x=-1

. (a) Briefly describe the shape of the region

R

and indicate the line

x=-1

on a sketch of the

xy

-plane. (b) For

0\le x\le4

, write expressions for the outer radius

R(x)

and the inner radius

r(x)

of a representative cross-section of the solid generated by the rotation. (c) Set up and evaluate a definite integral that gives the volume of the solid produced. Express your final answer in exact form.

Intuitive explanation

The key is recognizing this as a shell method problem, where revolving around a vertical line $x=-1$ (not the $y$ -axis) means each thin vertical strip traces out a cylindrical shell. The radius of each shell is the horizontal distance from the strip to the axis of rotation, and the height is determined by the curve $y=\sqrt{x}$ . Set up the integral by identifying the radius function and height function in terms of $x$ , then integrate across the region.

Intuitive explanation

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