The GATE Grind

GATE 2021 ME (ME2) – Question 36

Engineering Mathematics · Calculus: Definite and improper integrals, double and triple integrals · 2 marks · Multiple choice

The value of $\int_0^{\pi/2}\int_0^{\cos\theta}r\sin\theta\,dr\,d\theta$ is

  1. 0
  2. $1/6$
  3. $4/3$
  4. $\pi$

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Correct answer: (B) $1/6$

Explanation

Integrate over r first to obtain $\tfrac12\int_0^{\pi/2}\cos^2\theta\sin\theta\,d\theta$. Let u=cosθ, du=−sinθdθ.

The integral is $\tfrac12\int_0^1u^2du=\mathbf{1/6}$, **B**.