Compute the volume of B.

Question

Let B be the solid whose base is the circle x^2 + y^2 = 196 and whose vertical cross sections perpendicular to the x-axis are equilateral triangles. Compute the volume of B.

Answer 1

Volume, B = ∫_a^b A(x) dx

Where A(x) is the area of the cross section.

Area of an equilateral triangle, A = (√3/4)s², where s is the length of a side. A side is going to be

s=2√(196-x²). The base of B is a circle of radius of 14. Need to sum up all the triangle areas for x = -1 to 1.

A = (√3/4)(4)(196-x²) = √3(196-x²)

B = ∫_-1¹√3(196-x²)dx = √3[196x-x³/3]_-1¹ = √3(1174/3)