How to find the volume of a paraboloid using shells (y=1-x^2 revolved about the y-axis).

A paraboloid of revolution is defined by the curve y=1-x^2, which makes a region in the first quadrant bounded by the x and y axes. Revolving this region about the y-axis, we obtain a solid paraboloid, and we show how to find the volume of a paraboloid using shells.

We start by visualizing a thin cylindrical shell with an arbitrary radius of x. This shell has a height given by the height on the curve y=1-x^2. We cut and unroll the shell to create a thin rectangular slab, which will have the same height.

The length of the rectangular slab is given by the circumference of the original cylindrical shell, so that's 2pi*x, and finally the thickness of the shell is given by the infinitesimal increment in x, dx.

Now we can simply use length*height*thickness to get the volume of our rectangular solid, and that gives us the volume contribution of a single shell: dV=2pix(1-x^2)dx.

Finally, we use integration to add up all the volume increments, so we compute the integral of 2pix(1-x^2)dx on the interval [0,1] and we obtain the volume of the paraboloid defined by the curve y=1-x^2 revolved about the y-axis.