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Can someone tell me if my answer is right?

a) Evaluate the iterated integral $$\int_0 ^{\pi/2} \int_0 ^{\cos\theta} r^2 \sin\theta\, dr \,d\theta $$

My calculations

My answer is $0$.

Is this correct?

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1 Answer

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$$\int_{0}^{\frac{\pi}{2}}\int_{0}^{\cos \theta} r^2 \sin \theta \ dr = \int_{0}^{\frac{\pi}{2}}\sin \theta \int_{0}^{\cos \theta} r^2 \ dr = \frac{1}{3} \int_{0}^{\frac{\pi}{2}} \sin \theta\ r^3\bigr|_0^{\cos \theta} = \frac{1}{3} \int_{0}^{\frac{\pi}{2}}\sin \theta \cos^3 \theta \ d\theta$$

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Letting $u = \cos \theta$ we have $du = - \sin \theta$ i.e we have:

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$$-\frac{1}{3}\int_{1}^{0} u^3 \ du = \frac{1}{3} \int_{0}^1 u^3 \ du = \frac{1}{12} \cdot u^4 \bigr|_{0}^1 = \frac{1}{12}$$

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