Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = sqrt(25x) and y = x^2/25. Find V by slicing & find V by cylindrical shells.

Explanation:

Let and . The differential volume dV of the cylindrical shells is given by

Integrating this expression, we get

To determine the limits of integration, we equate the two functions to find their solutions and thus the limits:

We can clearly see that x = 0 is one of the solutions. For the other solution/limit, let's solve for x by first taking the square of the equation above:

or

Since we are rotating the functions around the y-axis, we are going to use the x = 25 solution as one of the limits. So the expression for the volume of revolution around the y-axis is