Finding Volume- Shells Method
1. Use desmos.com to plot your region.
2. Split your a to b interval into 6-8 equal subintervals.
3. Design and print a cylinder shell (meaning there is no thickness to the cylinder, but in reality
you will need to give it a little thickness or we won’t be able to see the object) at each x value
you split the interval into. The cylinder shell is created with a height of f(x) and a radius of x.
x
f(x)
4. Sketch the shape your graph would look like if you created more cylinders in the same way.
Then create and print this object.
5. Use the formula for circumference of a circle to find the general formula for the surface area
formula of a cylinder shell.
6. To find the volume of this shape you would add the surface areas of all of the cylinders
together, but we really want the exact volume. To find this, think of creating an infinite amount
of cylinder shells. We saw a similar situation when we first defined the integral as an infinite
𝑏
sum of rectangles. Similarly, the volume is ∫𝑎 2𝜋𝑥𝑓(𝑥)𝑑𝑥. Use this formula to find the volume
of your original graph rotating around the y-axis.