Question #8f07d

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Question #8f07d

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Answer 1 · 2017-03-27T15:30:25

$V = 36 π$ $V= 36pi$

Explanation:

Consider the line:

$y = \frac{x}{4}$ $y= x/4$

For $x = 12$ $x=12$ we have $y = 3$ $y=3$ , so we can see that rotating this line around the $x$ $x$ -axis in the interval $x \in (0, 12)$ $x in (0,12)$ generates exactly the solid requested.

The element of volume generated by the cylindrical shell between $x$ $x$ and $x + d x$ $x+dx$ is given by:

$d V = π y^{2} (x) d x = π \frac{x^{2}}{16} d x$ $dV = pi y^2(x)dx = pix^2/16dx$

so, integrating over the interval:

$V = \frac{π}{16} \int_{0}^{12} x^{2} d x = \frac{π}{16} {[\frac{x^{3}}{3}]}_{0}^{12} = 36 π$ $V= pi/16 int_0^12 x^2dx = pi/16 [x^3/3]_0^12 = 36pi$

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Question #8f07d

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