Thread: Need a method to find radii for Disk Method and h(x) and p(x) for Shell Method

To determine the volume of an object. I understand everything except how to figure out what the outer and inner radii are. My textbook does not give a clear method to determine what R(x) and r(x) should be for the Disk Method or what p(x) or (h)x should be for the Shell method. I have noticed there seems to be certain patterns and if there is just one curve and it is rotated around the x-axis, then it's just pi * the definite integral of R(x)^2 - r(x)^2. And the Shell method is 2 pi * times the definite integral of p(x) * (h)x). I just don't know how to find the radii for the Disk method and p(x) and h(x) for the Shell method. Online tutors that use the same textbook(calcchat.com) can't even figure it out either or at least explain it to me. When the axis of rotation is not the x or y axis, I can't figure it out a method to setup the integral(s). I have spent many hours trying to figure out a method to find how to do this with no luck, because the book sucks and I cannot find anything on the net that really explains it either!

thanks

Originally Posted by mojo0716

To determine the volume of an object. I understand everything except how to figure out what the outer and inner radii are. My textbook does not give a clear method to determine what R(x) and r(x) should be for the Disk Method or what p(x) or (h)x should be for the Shell method. I have noticed there seems to be certain patterns and if there is just one curve and it is rotated around the x-axis, then it's just pi * the definite integral of R(x)^2 - r(x)^2. And the Shell method is 2 pi * times the definite integral of p(x) * (h)x). I just don't know how to find the radii for the Disk method and p(x) and h(x) for the Shell method. Online tutors that use the same textbook(calcchat.com) can't even figure it out either or at least explain it to me. When the axis of rotation is not the x or y axis, I can't figure it out a method to setup the integral(s). I have spent many hours trying to figure out a method to find how to do this with no luck, because the book sucks and I cannot find anything on the net that really explains it either!

thanks

short and sweet: the inner radius is the smaller one and the outer radius is the larger one. that is, the inner radius is the smaller distance (from the axis of rotation to the closest curve to it) while the outer radius is the distance from the axis of rotation to the curve farther from it. if you have good spacial visualization, the outer radius comes from what would be the outside skin of the object formed. for example, rotating a rectangle around a vertical axis, you would get something that looks like a ring. the distance from the axis of rotation to the outside of the ring (the part that shows when someone wears it) is the outer radius, the distance from the axis of rotation to the inside (the part that touches the skin when you put the ring on) is the inner radius.

Originally Posted by Jhevon

short and sweet: the inner radius is the smaller one and the outer radius is the larger one. that is, the inner radius is the smaller distance (from the axis of rotation to the closest curve to it) while the outer radius is the distance from the axis of rotation to the curve farther from it. if you have good spacial visualization, the outer radius comes from what would be the outside skin of the object formed. for example, rotating a rectangle around a vertical axis, you would get something that looks like a ring. the distance from the axis of rotation to the outside of the ring (the part that shows when someone wears it) is the outer radius, the distance from the axis of rotation to the inside (the part that touches the skin when you put the ring on) is the inner radius.

Thanks. I'll see if I can figure these problems out with that advice.