Sketch the solid obtained by rotating the region underneath the graph of the function f(x)=x^(4) over the interval [0,1] about the axis x=2 and calculate its volume.
I'm starting to review for my Calculus II final and I am reviewing cylindrical shells and finding volume. I have started this problem and I'm think I have to use the shell method, but I'm not sure if I could use the Washer method also. I've started with the Shell method but something isn't adding up. I've gotten as far as:
∫ 2∏(2+x)*x^(4)dx when the limits are 0 to 1.
I've integrated it to 2∏((2x^(5)/5)+(x^(6)/6) but it doesn't seem correct. Does anybody know where I could've gone wrong?
Oh, I actually just realized what I did wrong, when I wrote out the integral I wrote ∫ 2∏(2+x)*x^(5)dx when the limits are 0 to 1. when I should have written ∫ 2∏(2-x)*x^(5)dx when the limits are 0 to 1. I then integrated it as 2∏((2x^(5)/5)-(x^(6)/6) and then after plugging the limits into it I came to an answer of (7∏/15). Does anyone know why the Washer Method can or cannot be used in this case? Would that just make it more difficult?