Volumes of Solids of Revolution

Abstract

This section is quite intuitive. A good spacial sense or drawing can help easily visualize the shape and pick the right formulas. Using area between two curves and knowing how the Riemann sum work are crucial.

  1. Figure out how you want to slice the solid
  2. Pick an arbitrary point and write out the area for a , and use
  3. Turn it into an integral.
Remark

There won't be any definitions in this note. Make sure to follow the logic.

How do we find the volume of these solids?

  1. Partition the axis in question into intervals of length (or )
  2. Make an approximation of the volume for each corresponding solid and add them up
  3. Take a limit as the width of the subintervals goes to zero

This is just the Riemann Sum and definite integral.

But how do we find the volume element?

Vertical Rectangle Revolved About a Horizontal Axis

Roughly, the slice is a cylinder, so the volume of that slice is: $$ \begin{align} \Delta V & \approx (\pi r^2)\Delta x \\ & \approx (\pi f(x)^2)\Delta x \\ dV & = \pi f(x)^2dx \end{align} $$ (see differential) Thus $\displaystyle V = \int \, dV = \int_{a}^b \pi f(x)^2 \, dx$
Example

In the above diagram, the curve over is revolved around the x-axis.

What if we rotate the curve about with ?

Now, instead of a solid disk in the centre, there's a hole. We need the Area Between Two Curves

Vertical Rectangle Revolved About a Vertical Axis

The volume element is obtained by rotating the vertical rectangle.

We have that and