Equipped with the integral and able to calculate it for many functions, we now move on to some interesting applications, each arising from the notion of a limit of sums. The integral was first introduced with reference to the "area under the graph" of a function. We begin this section by applying this application to more general regions in the plane.
This will allow us to move up from two dimensions to three, in order to calculate the volume contained within certain surfaces of revolution, a category of surfaces that includes spheres, cones, and cylinders. The integral will also enable us to compute the volume of solids given the cross-sectional areas perpendicular to an axis.
We continue by showing how the integral allows us to easily compute the average value of a function on a particular interval and even the length of its graph from one point to another.
We conclude our study of the basic applications of the integral by using it to find the total distance traveled by an object over a certain period of time when its velocity at each moment is known. This will highlight, once again, the crucial importance of the Fundamental Theorem of Calculus as the place where the derivative and integral are able to knock a few sparks off each other to illuminate the calculus landscape.