How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region $y = 2 - x$ $y=2-x$ , $2 \leq x \leq 4$ $2<=x<=4$ rotated about the x-axis?

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How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region $y = 2 - x$ $y=2-x$ , $2 \leq x \leq 4$ $2<=x<=4$ rotated about the x-axis?

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Answer 1 · 2015-10-09T17:40:44

Always graph the constraints first, then determine the correct integration formula.

Explanation:

Here is a graph of the constraints:

enter image source here

Now, without using calculus, you should visualize rotating this line around the x-axis will generate a cone with radius 2 and height 2. The volume of a cone is:

$V = (\frac{1}{3}) π r^{2} h = (\frac{1}{3}) π 2^{2} \cdot 2 = (\frac{8}{3}) π$ $V=(1/3)pir^2h=(1/3)pi2^2*2= (8/3)pi$

So, in advance of using calculus, we know the answer is $(\frac{8}{3}) π$ $(8/3)pi$

Now, using calculus and the shell method :

Because we rotate around the x-axis, integrate with respect to y:

$2 π \int_{- 2}^{0} y [4 - (2 - y)] d y = (\frac{8}{3}) π$ $2piint_-2^0y[4-(2-y)]dy=(8/3)pi$

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How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region $y = 2 - x$ $y=2-x$ , $2 \leq x \leq 4$ $2<=x<=4$ rotated about the x-axis?

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How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region y=2−xy=2-x, 2≤x≤42<=x<=4 rotated about the x-axis?

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Explanation:

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How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region $y = 2 - x$ $y=2-x$ , $2 \leq x \leq 4$ $2<=x<=4$ rotated about the x-axis?