Problem 31 - Essential Calculus

Question

Consider a solid that is generated by revolving a plane region about the \(y\) -axis. Describe the position of a representative rectangle when using (a) the shell method and (b) the disk method to find the volume of the solid.

Step-by-Step Solution

Answer

For the shell method, the rectangle is vertical and parallel to the \(y\)-axis. Its position is at a distance of \(x\) from the axis of rotation. For the disk method, the rectangle is horizontal and perpendicular to the \(y\)-axis. Its position is at a distance of \(g(y)\) from the axis of rotation.

Step 1: Shell method

The shell method involves using vertical rectangles to generate cylindrical shells. When a region is revolved about the \(y\)-axis using the shell method, the rectangle that is used is vertical and parallel to the axis of rotation - the \(y\)-axis. The height of the rectangle is a function of \(x\), \(y=f(x)\), and the width is an infinitesimally small change in \(x\), \(dx\). The rectangle is at a distance of \(x\) from the axis of rotation.

Step 2: Disk method

In contrast, the disk method involves using horizontal rectangles to generate cylindrical disks. When a region is revolved about the \(y\)-axis using the disk method, the rectangle is horizontal and perpendicular to the axis of rotation - the \(y\)-axis. The width of the rectangle is a function of \(y\), \(x=g(y)\), and the height is an infinitesimally small change in \(y\), \(dy\). The rectangle lies on a radius of the solid that is \(g(y)\) units from the axis of rotation.