Find the volume of the solid whose base is the circle x^2+y^2=64 and the cross sections perpendicular to the x-axis are triangles whose height and base are equal.

Question

Find the volume of the solid whose base is the circle x^2+y^2=64 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=7.

Steve · Answer

each triangle at a distance x from (0,0) has a base of 2y = 2√(64-x^2)

So, the total volume, taking advantage of symmetry is

v = 2∫[0,8] bh/2 dx
= 2∫[0,8] 2√(64-x^2)*2√(64-x^2)/2 dx
= 4∫[0,8] (64-x^2) dx
= 4096/3