Thread: Finding the volume of a function

1. 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.

thanks

2. 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.

3. 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, $y= \frac{1}{1+ x^2}$, 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 $2\pi x$. It's height is y so its surface area is $2\pi xy$ and, taking its thickness to be dx, its volume is $2\pi xy dx= 2\pi x\left(\frac{1}{1+ x^2}\right)dx= 2\pi \frac{xdx}{1+ x^2}$. To find the total volume, integrate that over all possible values of x: from 0 to 1.

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