Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve y = √(5x) from 0 to 5.

Answer 1

When a curve is rotated about the x axis, the radius of the pieces (shells are radius y above the x axis) with height 5-x = 5 - (y²/5) and thickness dy. You can verify that y(0) = 0 and y(5) = 5.

thus V =₀ ⁵∫2πydx =∫2πy(5 -y²/5)dy =2π ₀∫⁵ (5y - y³/5)dy = 2π[5y²/2 - y⁴/20]_o⁵ =2π(125/2 -625/20)=125π/2