# 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

Please answer letter c. The other letters have been answered already on enotes lately.

(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.

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### 1 Answer

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` .**

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