Volume of a circle $x^2 +y^2 \leq 1$ which is revolving around a line $x+y=2$.

Question

I want to compute the volume of a circle $x^2 +y^2 \leq 1$ which is revolving around a line $x+y=2$. Usually I solved problems about solids revolving around axis and non axis horizontal and vertical lines. But it seems to have neither horizontal nor vertical line. So I find it difficult for me to understand how can I solve this.

Can I get any help for this?

Answer 1

One approach is to use a change of coordinates. Rotate the graph $45^\circ$ (and redraw the axes) to obtain this equivalent problem: find the volume of the solid obtained by revolving the circle $x^2 + y^2 \leq 1$ about the line $y = \sqrt{2}$.

Answer 2

Using Pappus theorem, the volume of revolution is $2 \pi d A$, where $d$ is the distance from the centroid of the region to the axis of rotation and $A$ is the area bounded by the region. The centroid of the region is $(0,0)$ and the distance to the line $x+y-2=0$ is $\frac{|0+0-2|}{\sqrt{1^2+1^2}}=\sqrt{2}$, and $A$ is $\pi$. Therefore, the volume of revolution is $2\pi \sqrt{2} \pi$ or $\boxed{2\sqrt{2} \pi^2}$.