Q.271. Applications of integration, Volume of a solid of revolution.

Question 271.
The curves y = sin(x), y = cos(x) and x=0 bound a triangular shaped region. If the region is rotated about the y-axis, find the volume generated.

Answer 271.
sinx intersects y-axis at y = 0 and cosx intersects it at y = π/2
and the curves intersect at y = π/4
=> requird volume
= π∫(0 to π/4) sin^2 x dx + π∫(π/4 to π/2) cos^2 x dx
= π/2 [∫(0 to π/4) 2sin^2 x dx + ∫(π/4 to π/2) 2cos^2 x dx]
= π/2 [[∫(0 to π/4) (1 - cos2x) dx + ∫(π/4 to π/2) (1 + cos2x) dx]
= π/2 [(x - (1/2)sin2x) (0 to π/4) + (x + (1/2)sin2x) (π/4 to π/2)]
= π/2 [π/4 - (1/2) + π/4 - (1/2)]
= π (π/2 - 1).

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