I wanted to find the volume of a cylinder, radius r, height h by slicing it in to rectangles:
I placed the cylinder on the x-axis, one corner of the base diameter at (0,0) the opposite at (2r, 0). I have found that an area of a cross-section perpendicular to the x-axis is $A(x) = 2h \sqrt{r^2 -(x-r)^2}$ so:
$V=\int_0^{2r} 2h \sqrt{r^2 -(x-r)^2}dx$
I have tested this and it gives the volume correctly for various r and h. But how to show:
$\int_0^{2r} 2h \sqrt{r^2 -(x-r)^2}dx = \pi r^2 h$ without just saying "we know $V= \pi r^2 h$"?
This problem was my own device, perhaps it is not possible to do this.