Finding Volumes with Known Cross Section

Question

1. Let R be the region in the first quadrant bounded by the graph of y=2√x, the line y=6, and the y-axis. a) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid? b) The region R is the base of a solid. For this solid, each cross-section perpendicular to the y-axis is a semicircle. Write, but do not evaluate, an integral expression that gives the volume of the solid. 

Answer 1

a) Cross section area A(x) = (6 - 2√x)² and volume v = ∫₀⁹(6 - 2√x)²dx = (36x - 24 ·2/3x^3/2 + 2x²)₀⁹ = 54

b) Cross section area A(y) = π(y²/4)²/8 = πy⁴/128; volume v = ∫₀⁶πy⁴/128dy = πy⁵/640₀⁶ = 243π/20 = 38.17