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 $x=y^2$ , $y=0$ , and $y=sqr2$ 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 $x=y^2$ , $y=0$ , and $y=sqr2$ rotated about the x axis?

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Answer 1 · 2015-07-28T23:27:26

It's unclear what your solid is, but the only one that makes sense is $y = sqrt2$ and not $y = 0$ because $y = 0$ is redundant when you are rotating about the x-axis. Maybe you meant to emphasize that it is $y = sqrtx$ and not $y = -sqrtx$ . Assuming that:

$x = y^2 -> y = sqrtx$
You only need the positive one since $y = 0$ is a boundary, and is also your axis of rotation. Thus, you are looking at this:

http://www4a.wolframalpha.com/

Since the shell method implies you are rotating about the y-axis (which is inconvenient for the regular revolution method), we can rewrite this for that.

$x = y^2 -> y = x^2 in [0, oo)$ (only positive $x$ )
$y = 0 -> x = 0$
$y = sqrt2 -> x = sqrt2$

Now we really just have $y = x^2$ in the first quadrant rotated around the y-axis, stopped at $x = sqrt2$ , intersecting at $(sqrt2, 2)$ . Which is:

http://www4a.wolframalpha.com/

The shell method is:

$int_a^b 2pixf(x)dx$

where:
$2pi$ is the radian measure for the "circumference" of the shape
$x$ is the radius of the shell
$f(x)$ is the height of the shell

$V = 2piint_0^sqrt2 x(x^2)dx$

$= 2pi (x^4/4)|_(0)^sqrt2$

$= 2pi ((sqrt2)^4/4 - 0^4/4)$

$= 2pi (1)$

$= color(blue)(2pi "rad")$

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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 $x=y^2$ , $y=0$ , and $y=sqr2$ 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 x=y^2x=y^2, y=0y=0, and y=sqr2y=sqr2 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 $x=y^2$ , $y=0$ , and $y=sqr2$ rotated about the x axis?