Reverse the integration order in I= 01 0 ydx\,dy+ 12 y-11dx\,dy.

$\int_0^{1/2}\int_{-x}^xdy\,dx+\int_{1/2}^1\int_{-x^2}^{1-x^2}dy\,dx$

$\int_0^{1/2}\int_{x^2}^{x^2+1}dy\,dx+\int_{1/2}^1\int_{\sqrt x}^{\sqrt{x+1}}dy\,dx$

GATE 2026 MA – Question 12

Calculus · Multiple Integrals · 1 mark · Multiple choice

Reverse the integration order in $$I=\int_0^1\int_0^{\sqrt y}dx\,dy+\int_1^2\int_{\sqrt{y-1}}^1dx\,dy.$$

$\int_0^{1/2}\int_{-x}^xdy\,dx+\int_{1/2}^1\int_{-x^2}^{1-x^2}dy\,dx$
$\int_0^1\int_{x^2}^{x^2+1}dy\,dx$
$\int_0^{1/2}\int_{x^2}^{x^2+1}dy\,dx+\int_{1/2}^1\int_{\sqrt x}^{\sqrt{x+1}}dy\,dx$
$\int_0^1\int_{x^2}^{x^2-1}dy\,dx$

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Correct answer: (B) $\int_0^1\int_{x^2}^{x^2+1}dy\,dx$

Explanation

The first region has 0 ≤ x ≤ 1 and x² ≤ y ≤ 1. The second has 0 ≤ x ≤ 1 and 1 ≤ y ≤ x²+1. Their union is the vertical strip between y = x² and y = x²+1 for 0 ≤ x ≤ 1, giving **B**.

GATE 2026 MA – Question 12

Explanation

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