GATE 2020 ME (ME2) – Question 13

Engineering Mathematics · Calculus: Definite and improper integrals, double and triple integrals · 1 mark · Multiple choice

Let $I=\int_{x=0}^1\int_{y=0}^{x^2}xy^2\,dy\,dx$. Which expression also represents I?

$\int_{y=0}^1\int_{x=0}^{\sqrt y}xy^2\,dx\,dy$
$\int_{y=0}^1\int_{x=\sqrt y}^1yx^2\,dx\,dy$
$\int_{y=0}^1\int_{x=\sqrt y}^1xy^2\,dx\,dy$
$\int_{y=0}^1\int_{x=0}^{\sqrt y}yx^2\,dx\,dy$

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Correct answer: (C) $\int_{y=0}^1\int_{x=\sqrt y}^1xy^2\,dx\,dy$

Explanation

The region is 0≤x≤1,0≤y≤x². For a fixed y between0 and1, x ranges from √y to1. Preserve the integrand and reverse differentials: $\int_0^1\int_{\sqrt y}^1xy^2\,dx\,dy$. This is **C**.