The base of a solid is bounded by y =|x|+a, 0<a<3, and the line y=3. find in cu. units in terms of a, the volume of the solid if every cross section perpendicular to the y-axis is an equilateral triangle.

Answer 1

each cross-section has base 2x and altitude x√3

So, we want to add up all those triangles

v = ∫[a,3] 1/2 * 2x * x√3 dy
= √3 ∫[a,3] x^2 dy

But,
y = |x| + a
y-a |x|
(y-a)^2 = x^2

v = √3 ∫[a,3] (y-a)^2 dy

and now it's cake, right?