Finding the volumes of solids of revolution starts with rotating a shaded region around an axis to create different shapes like spheres, cones, and even squashes. While there are formulas for calculating volume of solids for known shapes, like spheres and cones, calculus is needed to find the exact volume of irregular shapes. By slicing the shape into infinitely thin slices and using integration, the exact volume of solids can be calculated.Let’s think about what happens when we take a shaded region and revolve - or rotate - it around a line or an axis, sweeping out everything we come in contact with as we go. This creates what is known as a “Solid of Revolution,” and we want to find the volume of solids like this.
Let’s rotate the region that’s bounded by this semi-circle around the x-axis. What solid would be get, and can we find its volume? Well, we’d get a ball, or sphere, right? And we might not have it memorized, but we could look up the formula that would give us the volume of the sphere. It’s 4/3 times pi times the radius of the sphere cubed.
Now, let’s look at another region.
Suppose we take the shaded region bounded by this linear function and rotate it around the x-axis, sweeping the area out as we go. This time we get a cone on its side, right? And if we wanted to find the volume of this solid, we’d look up the formula for the volume of a cone and use that. It’s pi/3 times the height of the cone times its radius.
Now, let’s try this – let’s take the shaded region bounded by this generic function, f(x), and rotate it around the x-axis. What solid do we get this time? Hmm…we get a butternut squash. But how are we going to find the volume of the squash? There just aren’t any formulas out there for a squash! So, what can we do?
Well, if we slice the squash up, each of the slices is going to be a cylinder. And we do have a formula to find the volume of a cylinder, right? It’s pi times the radius of the cylinder squared times its height, which in this case is the thickness of each slice since our cylinders are lying on their sides. Then if we add up the volume of these cylindrical slices, that would at least give us an approximation of the volume of the squash.
If we wanted a better approximation, we could slice the squash into thinner slices and add up their volumes. But this would still only ever give us an approximation of the volume of the squash. To find the exact volume of the squash, we need Calculus.
Calculus allows us to take this slicing idea and build on it to find the exact volume of the squash. With Calculus, it’s like we have an amazingly sharp knife. It’s so good that we can cut the squash into super, super thin slices. In fact, we can make them infinitesimally thin. So we are going to have infinitely many infinitesimally thin slices. And then we can use the process of integration to add up the volumes of these infinitely many super thin slices. The result is going to be the exact volume of the squash!