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 $y = 4 - x^{2}$ $y=4-x^2$ and $y = 0$ $y=0$ rotated about the y-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 $y = 4 - x^{2}$ $y=4-x^2$ and $y = 0$ $y=0$ rotated about the y-axis?

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Answer 1 · 2015-10-14T01:59:44

See the explanation, below.

Explanation:

Here is the region with a thin slice taken parallel to the axis of rotation. (To set up for cylindrical shells.)

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The volume of a representative shell is $2 π r h \cdot thickness$ $2pirh*"thickness"$

In this case, we have radius $r = x$ $r = x$ , height $h = 4 - x^{2}$ $h = 4-x^2$ and $thickness = d x$ $"thickness" = dx$ . $x$ $x$ varies from $0$ $0$ to $2$ $2$ , so the volume of the solid is:

$\int_{0}^{2} π x (4 - x^{2}) d x = π \int_{0}^{2} (4 x - x^{3}) d x = 4 π$ $int_0^2 pi x(4-x^2)dx = piint_0^2 (4x-x^3) dx = 4pi$

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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 $y = 4 - x^{2}$ $y=4-x^2$ and $y = 0$ $y=0$ rotated about the y-axis?

1 Answer

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=4−x2y=4-x^2 and y=0y=0 rotated about the y-axis?

1 Answer

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 $y = 4 - x^{2}$ $y=4-x^2$ and $y = 0$ $y=0$ rotated about the y-axis?