Find the volume of the solid obtained by rotating the region bounded by the given curves about the specifed line.

Question

Find the volume of the solid obtained by rotating the region

bounded by the given curves about the specifed line. Sketch the

region, the solid, and a typical disk or washer.

 

x=2sqrt(y), x= 0, y= 9; about the y- axis

Answer 1

To calculate the volume, you will need to use integration, but what integral and how?

We start with the curve itself which is defined by the equation x=2sqrt(y). For each value of y, x=2sqrt(y) will be the distance of the curve at that value of y from the y-axis.

If you rotate the curve around the y-axis, for each value y, you will trace a circle whose radius will be equal to r=2sqrt(y). The area of this circle will be πr². Now how do we get the volume traced by the rotating curve? We take very tiny slices of the volume composed of those circles I just described i.e. disks; the thickness of each disk will be dy, because we are going along the y-axis. The volumes (infinitesimal volumes to be exact) will be equal to the area of the disk times the thickness.

dV=πr²dy

The total volume will be V=π∫r²(y)dy (here I write r(y) to emphasize that r is a function of y)

r(y) is going to be 2sqrt(y).

The rest of the solution is to determine the limits of the integration, which are the lowest and highest values of y covering the range in this problem.

If you have any questions, please let me know.

We are taking a curve in an x-y plane and rotating it about the y-axis.

If you take a