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We have the functions $f(x) = 10-x^2$ and $g(x) = 2^{2x+2}$. The plane $V$ is bound by the curves of $f$ and $g$. Calculate the volume of:

  • The solid formed when $V$ rotates around the x-axis.

  • The solid formed when $V$ rotates around the line $y=10$.

My question is: Why would these be different? The plane $V$ keeps the same area, so why would it matter around which line you revolve it?

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    $\begingroup$ Is the distance from (the centroid of) $V$ to the lines the same? $\endgroup$ – David Mitra Jan 21 '14 at 21:47
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    $\begingroup$ Imagine a circle. Now rotate it around its diameter. You obtain a sphere with a well-known volume $4\pi r^3/3$. Now rotate this sphere around a very distant axis - you'll obtain a giant torus, it's volume will be, clearly, much bigger than the voolume of that sphere. $\endgroup$ – TZakrevskiy Jan 21 '14 at 21:49
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Mentally split the area into tiny pieces. A piece taht is close to one axis may be far away from the other. When rotating around an axis far far away, the tiny piece sweeps over a much larger volume than with a nearby axis.

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Let's go at a simpler question first, which will build up to yours. Suppose you have a single point and you want to know the length of the circle it traces out when you rotate this point around some axis. This obviously depends on the axis you choose. At one extreme, if the axis passes through the point itself, the point will trace out a circle of radius zero, which is zero length. On the other hand, if the axis is a $d$ meters away from the point at it's closest, the point will trace out a circle of radius $d$ meters and length $2\pi d$ meters.

Now, re-examine your question. Each point in $V$ is like the point in my example above and contributes a volume of $\sim2\pi d$ to the solid of revolution where $d$ is that point's distance to the axis of rotation. Clearly $d$ depends on the choice of axis, so the point's contribution to the total volume also depends on the choice of axis. To be specific to your question of the two axes being the $x$ and $y$ axes, only points on the line $x=y$ will be equidistant to both those axes. Most points in $V$ are different distances from the $x$ and $y$ axes. Unless $V$ were very symmetric (say, a circle centered at (1,1)), the total contributions from all it's points will be different depending on the axis chosen.

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