I keep getting large numbers for Volume using the Shell Method in Calculus. I follow the steps and can't get the right answer. Any tips?

Question

So, I've been following the process:

-Sketch the function

-Radius= x + a

-Height= f(x)

-Integrate 2pi int^b_a(x+a)f(x)dx

and I know how to invert for rotating around a horizontal line. As an example for how lost I am, I had to solve:

y= -x^2 + x +4 over [0,2] about the y-axis

The radius is x

The height is f(x)

V= 2pi [int^2_0(xf(x)dx)]

I got V= 92.15?????

Nowhere close to correct.

I'm pretty sure I'm integrating correctly; antiderivatives are fairly easy for me. Any idea as to what I'm missing??

Answer 1

v=2π*(-x⁴/4+x³/3+2x²) evaluated from x=0 to x=2, is 40π/3

does that match your equation?