An Ellipse centered at the origin is described by the equation (x2/a2)+(y2/b2)= 1. If an ellipse R is revolved about either axis, the resulting solid is called an ellipsoid. Find the volume of the ellipsoid generated when R is revolved about the x axis.

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To calculate the volume of a curve y(x) around the x-axis, we use the following formula:

`V=int A(x)dx`

where A(x) denotes the cross sectional area at x and it is calculated via:

`A(x)=piy^2(x)`

To visualize this, imagine the ellipsoid laying on the x-axis. Now take out a cross sectional slice from it (like cutting a veeeeery thin slice of bread). The area of this slice is given by A(x). Note that it is very similar to the area of a circle (as if we were cutting a cylinder), but with a non-constant radius, given by y(x). What remains to be done is add those slices to get the volume of the solid, so we have to integrate the slices:

`V=int piy^2(x)dx `

but `y^2(x) ` is given by our equation for the ellipse:

` y^2(x)=1-x^2/a^2`

so we substitute it and we have the following integral:

`int_-a^a pi b^2(1-x^2/a^2)dx`

Note that we integrate from the extremal values of x, that is, from -a to a (the limits of our ellipse in the x-axis). Finally, we have:

`V=int_-a^a pi b^2(1-x^2/a^2)dx = pib^2(int_-a^a dx - int_-a^a x^2/a^2)) = pi b^2(2a-2/3a^3/a^2)=4/3 piab^2`

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