Find the volumes of the solids whose bases are bounded by the circle x^2 + y^2 = 16, with the indicated cross sections taken perpendicular to the x-axis.

Question

a. squares

b. equilateral triangle

c. semicircle

d. Isosceles right triangle

Answer 1

a. Volume v = ∫_-4⁴4(16 - x²)dx = 8(16x - x³/3)₀⁴ = 8(64 - 64/3) = 1024/3.

b. Volume v = ∫_-4⁴√3(16 - x²)dx = 2√3(16x - x³/3)₀⁴ = 2√3(64 - 64/3) = 256√3/3.

c. Volume v = ∫_-4⁴π/2(16 - x²)dx = π(16x - x³/3)₀⁴ = 128π/3

d. Volume v = ⌈_-4⁴2(16 - x²)dx = 4(16x - x³/3)₀⁴ = 512/3