# Using Integrals to Solve Applied Problems

If a curve is rotated about the *x* or *y* axis or a line parallel to one of the axes, to form a 3D object, we can calculate the volume of this solid by using the tools of integration. For example, let's say the parabola *y = x*2 is rotated around its axis of symmetry to form a paraboloid, as in *Figure 10.16*:

We can find the volume by adding up all the *slices* of the paraboloid as you go up the solid. Just as before, when we were using rectangles in two dimensions, now we're using cylinders in three dimensions. In *Figure 10.16*, the slices are going up the figure and not to the right, so we can flip it in our heads and redefine the curve *y = x*2 as *y = sqrt(x)*.

Now the radius of each cylinder is the *y* value, and let's say we're going from *x = 0* to *x = 1*:

The endpoints are still *0...*