Question

A solid has a base bounded by x^2_y^2=36. Find the volume of the solid if every plane section perpendicular to the diameter is an isosceles triangle whose base is on the circle and whose height is 4 units

Answer 1

since the area of a triangle is 1/2 base * height, if the base is 2y, then we need only double the volume of half the solid, due to symmetry.

v = 2∫[0,6] (1/2)(2√(36-x^2))(4) dx
= 8∫[0,6] √(36-x^2) dx

Now just plug and chug.