y = x^(1/3) - 2
Find volume across the x-axis on 8 <= x <= 27 using the Method of Shells
V = 2pi * Integral(r * h) from a -> b
The Attempt at a Solution
Let x = (y + 2)^3
Let h = (y +2)^3 // Set height
Let r = y // Set radius
2pi * Integral(y(y+3)^3 dy) from 0 to 1 // Integrate in terms of y
I multiplied (y+3)^3 out, and combined with y for:
2pi * Integral(y^4 + 6y^3 + 12y^2 + 8y dy) from 0 to 1
2pi ((y^5)/5 + ((3/2)*y^4) + (4y^3) + (4y^2)) from 0 to 1
I then calculate F(1) - F(0) and get (194/10)pi = (97/5)pi.
However, the correct answer in the book is (38/5)pi.
I also tried u-substitution and still got (97/5)pi.
Please help... this is extremely frustrating for me. I have been working really hard at finding volumes for the last two weeks and still continue to get most of my answers wrong. I've been to tutoring, my professor's office hours multiple times, and now I'm here. I just don't see what I'm doing wrong. I have an exam tomorrow morning and feel like I'm going to flunk it even though I've put in more than my fair share of studying.
I did set the radius and height correctly, right? If so, what is going on?