how do you find the solid of revolution if it doesn't rotate on either the x-or y-axis?

Question

are they the same for both disk and shell method?

thanks

Answer 1

Good question.

The underlying volume and the corresponding mathematics doesn't change with the axis of revolution. It is just that when the axis of revolution is parallel to either the X or Y axis, the computation of the integral is easier.

So, one option is to transform the more difficult integration into an easier one. (In one sense, that's what all the various techniques such as substitution and integration by parts, etc. are all about.)

In particular, a simple rotation in the plane will transform any axis to one parallel to a primary axis.

For example, consider the donut that is produced by rotating the circle (x-3)^2 + y^2 = 1 around the axis x = y.

And consider the 45 degree rotation induced by the transformation rotation) = T where T =

(cos 45) (sin 45)
(sin -45) (cos 45)

This moves the center of the circle from <3, 0> to <3 cos 45, 3 cos 45> and the axis of rotation to X = 0 (i.e. the Y axis)

Then we have = T' where T' is the inverse of T which just happens to be the rotation of 45 degrees in the opposite direction or T' =

(cos 45) (sin -45)
(sin 45) (cos 45)

or:
x = X(cos 45) + Y(sin -45)
y = X(sin 45) + Y(cos 45)

So now the equation of the circle becomes:

(X(cos 45) + Y(sin -45) - 3)^2 + (X(sin 45) + Y(cos 45))^2 = 1

with, as noted, the axis of rotation being the Y axis.

In this example, by design, the rotated equation is as easy to integrate as the original equation was. In general, this will not be the case. But then, the kinds of problems one encounters in homework and on exams aren't the general case either. (Most real integration problems can't be done symbolically, but only numerically. :-)