Different axes of revolution Suppose R is the region bounded b...

Question

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).

b. How is this formula changed if x0>b?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says suppose T is a region bounded by Y is equal to P X and Y is equal to Q of X on the closed interval from M to N, with PX being greater than or equal to Q X. If this region is revolved about the vertical line X2, where x2 is greater than N, what is the formula for the volume of the solid generated? Now, for this problem, we need to recall the general formula for the volume using the shell method. And here in our case, we have an axis of revolution that is vertical and outside the region. So recall that our volume B is going to be equal to. 2 i multiplied by the interval from M to N, since that is the interval that we're interested in for this problem. And for Ingram, we're going to have the radius. Of our show. And then that gets multiplied by the height. Of our shell And we're going to be integrating with respect to X, so we'll need to multiply this by DX. So, we just need to determine the radius and height of our shell. So, first, One that we're going to look at is going to be the radius of our shell, and our radius is just going to be the distance from our axis of rotation to a point X. And since our axis of rotation is located at X2, that means the radius is just going to be equal to X2 minus X. Next, we're going to look at the height. And the height here is just going to be the difference between our two Y curves. And since we're told that PF X is greater than Q of X in the problem, that means their height is going to be P X Minus Q X since we want our height to be a positive quantity. Now, we substitute these results back into our formula for our volume, and we see that our volume B is going to be equal to 2 i. Multiplied by the integral from M to N. Of the quantity of X2 minus X in quantity, multiplied by the quantity of PX. Minus Q X In quantity DX. So this here is the answer that we're looking for for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).

b. How is this formula changed if x0>b?

Key Concepts

Volume of Revolution

Definite Integrals

Axis of Revolution

Watch next