# Shell Method Problem - about x axis - # 2

#### Jason76

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

$$\displaystyle x = 14,$$

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

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

$$\displaystyle V = 288 * 2\pi = 576\pi$$ ???

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#### skeeter

MHF Helper
$$\displaystyle V = 2\pi \int_3^9 y \left(14 - [5 + (y-6)^2] \right) \, dy = 432 \pi$$