Question

Consider the ellipse x^2+4y^2=1 in the xy plane.

a. Of this ellipse is revolved about the​ x-axis, what is the equation of the resulting​ ellipsoid?
b. If this ellipse is revolved about the​ y-axis, what is the equation of the resulting​ ellipsoid?

Answer 1

a. x² +4y² +4z² = 1

b. x² +4y² +z² = 1

Explanation:

a. In "intercept form", the given equation can be written as ...

(x/1)² + (y/(1/2))² = 1

When the ellipse is revolved around the x-axis, the z-intercept will match the y-intercept, so the equation becomes ...

(x/1)² + (y/(1/2))² + (z/(1/2))² = 1

x² +4y² +4z² = 1

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b. Likewise, revolving the figure around the y-axis will make the z-intercept the same as the x-intercept. That equation is ...

x² + 4y² +z² = 1

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See the attachments. The first shows revolution about the x-axis; the second shows revolution about the y-axis.