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Thread: Finding the volume of a function

  1. #1
    Junior Member
    Sep 2010

    Finding the volume of a function

    Find the volume of the solid formed by rotating the region in the first quadrant bounded by y = 1/(1+x^2) and x = 1 about the y-axis.

    I know the formulas associated with finding volumes of a function, but I'm confused about this one.

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  2. #2
    Senior Member Educated's Avatar
    Aug 2010
    New Zealand
    When it is rotated around the y-axis, half of the volume will be in quadrant 1 and the other half will be in quadrant 2. All you have to do is halve the volume that you have calculated.
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  3. #3
    MHF Contributor

    Apr 2005
    Did he say that he had calculated any volume?

    In any case, it makes no sense to say that a volume is in a quadrant. The four "quadrants" are in the plane. In three dimensions, there are 8 "octants". Educated, you are misunderstanding the problem. It is only the two dimensional region that is bounded by x=0, y= 0, $\displaystyle y= \frac{1}{1+ x^2}$, and x= 1 that is in the first quadrant. it is then rotated around the y-axis so the volume occupies the first four octants.

    ikurwae89, draw a picture of the two-dimensional figure. Since you are given y as a function of x (and it would be a bit complicated to solve for x) but rotating around the y-axis, probably the simplest thing to do is to use "shells". That is, at each x, imagine a vertical line through the region to be rotated. As it rotates around, it will sweep out a cylinder. The radius of that cylinder will be x so its circumference is $\displaystyle 2\pi x$. It's height is y so its surface area is $\displaystyle 2\pi xy$ and, taking its thickness to be dx, its volume is $\displaystyle 2\pi xy dx= 2\pi x\left(\frac{1}{1+ x^2}\right)dx= 2\pi \frac{xdx}{1+ x^2}$. To find the total volume, integrate that over all possible values of x: from 0 to 1.

    The volume is $\displaystyle \int_0^1\frac{xdx}{1+x^2}$. That should be easy (let $\displaystyle u= 1+ x^2$).
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