Thread: Volume of a Solid of Revolution...?

1. Volume of a Solid of Revolution...?

Find the volume of the solid that is generated when the region in the first quadrant bounded by y=x2, y=4, and x=0 is revolved about the line y=-2.

I cannot figure out how to do these problems where the area is revolved around something other than the x or y axis. I am following the directions written in my book, but I must be misinterpreting something.

Help?

2. Re: Volume of a Solid of Revolution...?

Why not try moving the entire functions up by 2 units so that you can revolve it around the x axis?

3. Re: Volume of a Solid of Revolution...?

When a curve is revolved around an axis, each point on the curve moves in a circle having the point on the axis as center. The radius of that circle is the (perpendicular) distance from the point to the axis. Here, the two curves are $y= x^2$ and y= 4. The distance from a point $(x, x^2)$, on $y= x^2$, to (x, -4), on y= -4, is $x^2- 4$. The distance from a point (x, 4), on y= 4, to (x, -4), on y= -4, is 4- (-4)= 8.

So rotating a point on y= 4 around the axis y= -4 gives a circle of radius 8 and so an area of $64\pi$. Rotating a point on $y= x^2$ around the axis y= -4 gives a circle of radius $x^2- 4$ and so an area of $\pi(x^2- 4)^2$.

In other words, it is just a matter of extending the radius from the point to the given axis. As long as the axis is parallel to the x or y axis, that is relatively easy. Problems requiring rotation about an axis at an angle to the coordinate axes is much harder and you should not see those in an introductory calculus class.

4. Re: Volume of a Solid of Revolution...?

None of this worked for me. The answer is given as 704pi/3.