Volume of an ellipsoid Suppose that the ellipse \(x^2/a^2 + y^2/b^2 = 1\) is revolved about the x-axis. What is the volume of the ellipsoid that is generated? Is the volume different if the same ellipse is revolved about the y-axis?

Solution 79E

Step 1 :

Given : .

a. To calculate the volume of the ellipsoid, we know that the volume of the solid generated by revolving the curve y=f(x) about the x-axis is given as follows

Here the volume generated by is calculated by putting and using the limit from (-a,a) as follows

Hence the volume generated by revolving the ellipse about the x-axis is .

b. To calculate the volume of the solid generated by revolving the curve about the y-axis is given as follows

Here the volume generated by is calculated by putting and using the limit from (-b,b) as follows

Hence the volume generated by the ellipse revolved around the y-axis is .

Yes, the volume is different if ellipse is revolved about y-axis.