Shell Method: y=4(x-2)^2, and x^2-4x+7, about y-axis

[TwistedNerve]

Heres two problems.

1. Find the volume generated by rotating the region around the y-axis. \(\displaystyle y=4(x-2)^2\) and \(\displaystyle x^2 -4x+7\).

Im not sure why this one is giving me such a hard time. The region bounded by the curves is the integral of \(\displaystyle (x^2 -4x+7)-(4(x-2)^2)\) times the radius (all multiplied by 2 pi), but I wasnt sure what the radius was. My book says the answer is 16 pi but trying numerosu values for the radius I never got that answer.

2) Set up (but not evaluate) an integral that would find the volume generated by reflecting the area enclosed over the x-axis using the shell method.

The region is bounded by x=0, y=2, and \(\displaystyle y=e^x\) Since the shell method is being used and its being reflected over the x-axis I knew id be integrating with respect to y. I thought the radius was 1+y (???) so would the integral be \(\displaystyle \int_0^2(1+y)(ln y) dy\)??

Thanks so much you all are most helpful !!

 

[galactus]

#1. \(\displaystyle 2{\pi}\int_{1}^{3}\left[x(x^{2}-4x+7-4(x-2)^{2})\right]dx\)