Question

Use the shell method to find the volume of the solid generated by revolving the shaded region about y-axis. x=5, y=(5x)^1/2

Answer 1

Answer:

200π

Step-by-step explanation:

Graph: desmos.com/calculator/pwa9vqzmpy

The region is inside a horizontal parabola y = (5x)^½, starting at its vertex at x=0 and ending at x=5.  Cut a thin vertical slice with thickness dx and position x.

Revolving this thin slice about the y-axis, we get a cylindrical shell with radius x, thickness dx, and height 2y.  The volume of the shell is:

dV = 2π (x) (2y) (dx)

dV = 4π xy dx

dV = 4π x (5x)^½ dx

dV = 4π√5 (x^³/₂) dx

The total volume is the sum of all the shell volumes from x=0 to x=5.

V = ∫₀⁵ dV

V = ∫₀⁵ 4π√5 (x^³/₂) dx

Evaluating the integral:

V = 4π√5 ∫₀⁵ (x^³/₂) dx

V = 4π√5 (⅖ x^⁵/₂) |₀⁵

V = 4π√5 [(⅖ 5^⁵/₂) − (⅖ 0^⁵/₂)]

V = 200π

The volume of the solid is 200π cubic units.