Find the volume of the solid whose base is the circle x2+y2=25 and the cross sections perpendicular to the x-axis are triangles whose height and base are equal. find the area of the vertical cross section a at the level x=4.

Triangles with height and base , with have area .

Such cross sections with the base of the triangle in the disk (a disk with radius 5) have base with length

i.e. the vertical (in the plane) distance between the top and bottom curves describing the circle .

So when , the cross section at that point has base

so that the area of the cross section would be 6^2/2 = 18.

In case it's relevant, the entire solid would have volume given by the integral