Shell Method Problem - about x axis - # 2

Oct 2012
1,314
21
USA
Find Volume about x axis

\(\displaystyle x = 5 + (y-6)^{2}\)

\(\displaystyle x = 14, \)

Simplify

\(\displaystyle x = 5 + (y-6)^{2}\)

\(\displaystyle x = 5 + (y-6)(y-6) \)

\(\displaystyle x = 5 + y^{2} - 6y-6y + 36 \)

\(\displaystyle x = 5 + y^{2} - 12y + 36 + 5 \)

\(\displaystyle x = y^{2} - 12y + 41 \)

Find limits of integration

\(\displaystyle y^{2} - 12y + 41 = 14 \)

\(\displaystyle y^{2} - 12y + 27 = 0 \)

\(\displaystyle (y - 9)(y - 3) = 0 \)

\(\displaystyle y = 9, y = 3 \)

\(\displaystyle V = 2\pi \int_{3}^{9} (y)(y^{2} - 12y + 41) dy \)

\(\displaystyle V = 2\pi \int_{3}^{9} y^{3} - 12y^{2} + 41y dy \)


\(\displaystyle V = (2\pi) \dfrac{ y^{4}}{ 4} - \dfrac{ 12y^{3}}{3 } - \dfrac{41y^{2} }{ 2} \) evaluated at 3 and 9

\(\displaystyle V = (2\pi) \dfrac{ y^{4}}{ 4} - 4y^{3} - \dfrac{41y^{2} }{ 2} \) evaluated at 3 and 9

\(\displaystyle V = (2\pi) [[\dfrac{ (9)^{4}}{ 4} - 4(9)^{3} - \dfrac{41(9)^{2} }{ 2}] - [\dfrac{ (3)^{4}}{ 4} - 4(3)^{3} - \dfrac{41(3)^{2} }{ 2}]] \)

\(\displaystyle V = 288 * 2\pi = 576\pi\) ???
 
Last edited:

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
\(\displaystyle V = 2\pi \int_3^9 y \left(14 - [5 + (y-6)^2] \right) \, dy = 432 \pi\)