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.

Short 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-by-step solution

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.

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.