GATE 2022 EC – Question 53
The value of the integral
$$\iint_D 3(x^2+y^2)\,dx\,dy,$$
where $D$ is the shaded triangular region shown in the diagram, is ________ (rounded off to the nearest integer).
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Correct answer: 512
Explanation
The triangle has vertices $(0,0)$, $(4,4)$ and $(4,-4)$, so $0\le x\le4$ and $-x\le y\le x$. Then $\int_{-x}^{x}3(x^2+y^2)\,dy=3\left(2x^3+\frac{2x^3}{3}\right)=8x^3$, and $\int_0^48x^3\,dx=2x^4\Big|_0^4=512$.