Respuesta :
Using shell method, the volume of the solid generated by revolving the shaded region about the line y is 200π cubic units.
What is shell method?
A solid of revolution is divided into cylindrical shells and its volume is calculated using the shell method. We cut the solid perpendicular to the axis of revolution that produces the shells. The volume of the cylindrical shell is calculated by multiplying the cylinder's surface area by the thickness of the cylindrical wall.
The Shell Method Formula: What Is It?
Let R be the area enclosed by the lines x = a and x = b.
Consider the case when we rotate a solid about a vertical axis to create a solid. Let h(x) be the height of the shell and r(x) be the separation between the rotational axis and x.
V=2bar(x)h(x)dx is a formula for calculating the solid's volume.
How to solve?
The area is enclosed by a horizontal parabola with the equation y = (5x)12, which has its vertex at x=0 and its extremum at x=5. Make a narrow vertical slice with thickness dx and location x.
This thin slice may be rotated about the y-axis to create a cylindrical shell with the dimensions x, dx, and 2y. The shell's volume is as follows:
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
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