Question

Consider the region bounded by the graphs of x=y^2 and x=9. Find the volume of the solid that has this region as its base if every cross section by a plane perpendicular to the x-axis has the shape of an equilateral triangle.

Answer 1

so, the sides of each triangle have length 2x,
making the area of the triangle √3/4 (2x)^2 = √3 x^2
The volume is thus
∫[0,9] √3 x^2 dx = _____

Answer 2

oops. the side length is 2y, not 2x.
v = ∫[0,9] √3 y^2 dx = ∫[0,9] √3 x dx