PUNE MCA Exam 2015 Question Paper with answers for online practice

Concept:The volume is the double integral of the ceiling height function over the rectangular base.Explanation:Volume $V = \int_{y=0}^{12} \int_{x=0}^{12} (24 + y \sin y \cos x) \, dx \, dy$.Integrate with respect to $x$: $\int_0^{12} 24 \, dx = 288$, and $\int_0^{12} y \sin y \cos x \, dx = y \sin y \sin x \big|_0^{12} = y \sin y \sin 12$.So inner integral is $288 + y \sin y \sin 12$.Then $V = \int_0^{12} 288 \, dy + \sin 12 \int_0^{12} y \sin y \, dy = 288 \cdot 12 + \sin 12 \cdot I$, where $I = \int_0^{12} y \sin y \, dy$.$288 \cdot 12 = 3456$.Using integration by parts: $I = [-y \cos y + \sin y]_0^{12} = -12 \cos 12 + \sin 12$.Thus $V = 3456 + \sin 12 (\sin 12 - 12 \cos 12) = 3456 + \sin^2 12 - 12 \sin 12 \cos 12$.Evaluate numerically: $12$ radians $\approx 5.717$ rad, $\sin 12 \approx -0.5366$, $\cos 12 \approx 0.8440$, so $\sin^2 12 \approx 0.2879$, $-12 \sin 12 \cos 12 \approx 5.4324$, total extra $\approx 5.7203$. Hence $V \approx 3461.72$ cubic feet.This is between 3456 and 4000.Answer:The volume is approximately 3462 cubic feet, which is ≥ 3456 but less than 4000.