Definite integrals were originally introduced as a tool for finding "area," but with a small change in the same idea, they can also find the volume of a solid of revolution and the distance traveled. Explore these two situations to see how the core idea of slicing into thin pieces and adding them extends to other quantities.
Volume principle: Slice the solid of revolution into many thin disks, find each disk's volume (area × thickness), and add them all to get the total volume. If the radius of a disk is f(x), its area is π·f(x)², and integrating from x=0 to a gives V = π∫f(x)²dx.
Distance principle: Integrating velocity v(t) with respect to time gives the change in position (displacement). On a v-t graph, the signed area under the curve represents displacement. It is exactly the same idea as definite integration being about "area" in the first place—the horizontal axis is time and the vertical axis is velocity this time.