When you slice vertically, the top curve is y = x and the limits of integration are from x = 0 to x = 1. Using
our formula, we evaluate the integral.
π x^2 dxThe result is
π x^2 dx = π = By the way, did you notice that the solid in the problem is a cone with a height and radius of 1? The
formula for the volume of a cone is π r^2 h, so you should expect to get .
Now let’s figure out how to find the volume of the solid that results when we revolve a region that does
not touch the x-axis. Consider the region bounded above by the curve y = x^3 and below by the curve y =
x^2 , from x = 2 to x = 4, which is revolved about the x-axis. Sketch the region first.