# An ellipse centered at the origin is described by the equation ((x^2)/(a^2))+ ((y^2)/(b^2)) =1    Please answer letter c. The other letters have been answered already on enotes lately. (a) Set...

An ellipse centered at the origin is described by the equation ((x^2)/(a^2))+ ((y^2)/(b^2)) =1

(a) Set up an integral for the volume of the ellipsoid generated when the region defined by ((x^2)/(a^2))+ ((y^2)/(b^2)) =1 is revolved about the x-axis. Do not evaluate the integral.

b) Set up an integral for the volume of the ellipsoid generated when the region de ned by ((x^2)/(a^2))+ ((y^2)/(b^2)) =1 is revolved about the y-axis. Do not evaluate the integral.

(c) Should the results of (a) and (b) agree? Explain.

justaguide | Certified Educator

When an ellipse `x^2/a^2 + y^2/b^2 = 1` is revolved about either of the axes a 3-dimensional form called an ellipsoid is created.

The volume of an ellipsoid is `(4/3)*pi*a*b*c` where a, b and c are the length of the semi-principal axes along the three dimensions.

If an ellipse `x^2/a^2 + y^2/b^2 = 1` is revolved about the x-axis, the length of the semi-principal axes of the resulting ellipsoid a, b, b. This gives the volume of the ellipse as `(4/3)*pi*a*b^2`

On the other hand if the ellipse is revolved about the y-axis, the length of the semi-principal axes of the resulting ellipsoid is a, a, b. This gives the volume of the ellipse as `(4/3)*pi*a^2*b`

The volume obtained in the two cases is not the same as `a != b` .