I think the shell method uses the formula as follows However

I think the shell method uses the formula as follows:

However, I do not know for certain what to plug in for each variable in this problem.

I set

The first x is the radius function, which is just x. f(x) is the height function, and that is where I am lost, how do I find the height function for this problem?

Please give me a step by step walk-through for this problem, Thanks a lot. I will rate 5 stars if it helps me!

I think the shell method uses the formula as follows: 2Pi Integral (x)(f(x))(dx) between the limits a and b 3?7 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. 7.y=x^2, y=6x-2x^2 However, I do not know for certain what to plug in for each variable in this problem. I set X^2=6X-2X^2 and found that x=0 and x=6. I used these as my bounds of integration. The first x is the radius function, which is just x. f(x) is the height function, and that is where I am lost, how do I find the height function for this problem? Please give me a step by step walk-through for this problem, Thanks a lot. I will rate 5 stars if it helps me!

Solution

x^2 = 6x - 2x^2
3x^2 = 6x
x = 0, 2
f(x) = x^2 - (6x - 2x^2)
f(x) = 3x^2 - 6x
V = integral of 2xf(x)dx........limit from 0 to 2
V = integral of 2x(3x^2 - 6x)dx
V = integral of 2(3x^3 - 6x^2)dx
V = 2(3x^4/4 - 2x^3)
apply limits
V = 2(12 - 16)
V = 8