Q.157. Volume of solid of revolution.

Question 157.
Let R be the region that is bounded by the triangle with the vertices (0, 0), (2, 0), and (1, 1).
Find the volume of the solid generated by revolving the region R about the line x = 2.

Answer 157.
Let the given points be A(0, 0), B(2, 0) and C(1, 1)
By symmetry, the volume generated by revolving around x = 2 will be the same as generated by revolving around x = 0 (y-axis).
The equation of the line AC is y = x and of the line BC is y = 2 - x
=> required volume
= volume generated by revolving area under line y = 2 - x less volume generated by revolving area under line y= x between y = 0 and y = 1.
= π ∫ [(2 - y)^2 - y^2] dy ... (y=0 to 1)
= π ∫ (4 - 4y) dy ... (y=0 to 1)
= π (4y - 2y^2)
Plugging the linits 0 to 1
= 2π cubic units.

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