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.
