i need help with rotations? how do you determine what your equation to integrate is and what your boundaries are (if left out) and solve the problem? For example: Determine the volume of the solid obtained by rotating the region bounded by 2*(square root of x-1) and about the line x=6.

Question

i need help with rotations? how do you determine what your equation to integrate is and what your boundaries are (if left out) and solve the problem? For example: Determine the volume of the solid obtained by rotating the region bounded by 2*(square root of x-1) and  about the line x=6.

anonymous · Answer

Do you have a second bound, or is x=6 the second one?

anonymous · Answer

thats all we were given for the bounded region

anonymous · Answer

If I understand the question correctly, the method of cylindrical shells is the easiest way to go.  You have to set up an elemental cylindrical shell volume and integrate each shell.

The volume of a shell is:

dV = 2pi*radius*height*(elemental thickness)
dV = 2pi*(6-x)*y*dx
dV = 2pi*(6-x)*2sqrt(x-1)dx = 4pi(6-x)sqrt(x-1)dx

anonymous · Answer

If you expand:

dV = 4pi[6sqrt(x-1)-xsqrt(x-1)]dx and use the substitution u^2=x-1 in the second one, you should be able to pump it out.

anonymous · Answer

I have a class soon, so I hope this is enough to help you.

anonymous · Answer

You would be integrating from x = 1 to 6.

anonymous · Answer

thanks but how to you determine your intervals?

anonymous · Answer

Plot the function 2sqrt(x-1) from x=1 to 6.  You can't go below 1 since the radical won't be real anymore, and you can't go above 6 because it's the axis of rotation.