Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded on the right by the graph of g(y)=9√y and on the left by the y-axis for 0 ≤ y ≤3, about the x-axis.

Answer 1

Your graph is bounded between y=0 and y=3, and we'll be rotating the shells around the x axis, so that makes the radius equal to y. Then the volume of each shell is the area of the rectangle formed by the width of dy and the length of 9√y and the circumference of 2πy. This makes the overall integral be:

∫₀³18πy^1.5dy = 7.2π(3^2.5) = 64.8π√3 cubic units or 352.6026 cubic units.

Answer 2

x = 9√y; y = x²/81; 0 ≤ x ≤ 9√3.3

Volume v = π∫₀^9√3(3² - x⁴/81²)dx = π(9x - x⁵/32805)₀^9√3 = 352.6