GATE 2022 EC Q.53: solution and answer

GATE 2022 · ECQ.53 · 2 marks · NAT

The value of the integral

\iint_{D} 3 (x^{2} + y^{2}) d x d y,

where $D$ is the shaded triangular region shown in the diagram, is ______ (rounded off to the nearest integer).

GATE 2022 EC question 53: Cartesian axes: a horizontal x-axis and a vertical y-axis, each with arrows at both ends. — Figure from the GATE 2022 EC paper. It is part of the question.

Answer and solution

ANSWER

$\iint_{D} 3 (x^{2} + y^{2}) d x d y = 512$

Official answer: 512Source: Official answer key, GATE 2022 EC (IIT Kharagpur)

Describe the region. The triangle has vertices $(0, 0)$ , $(4, 4)$ and $(4, - 4)$ . The line $y = x$ bounds it above, the line $y = - x$ bounds it below, and the line $x = 4$ closes it on the right. Cut it into vertical strips: for each $x$ with $0 \leq x \leq 4$ , $y$ runs from $- x$ to $x$ .
Set up the iterated integral. The integrand is continuous on this closed, bounded region, so by Fubini's theorem we can integrate over $y$ first: $I = \int_{0}^{4} \int_{- x}^{x} 3 (x^{2} + y^{2}) d y d x$
Inner integral in $y$ . The integrand $x^{2} + y^{2}$ is even in $y$ and the limits are symmetric, so $\int_{- x}^{x} (x^{2} + y^{2}) d y = 2 \int_{0}^{x} (x^{2} + y^{2}) d y = 2 (x^{2} \cdot x + \frac{x ^{3}}{3}) = \frac{8 x ^{3}}{3}$
Multiply by the constant 3. $3 \cdot \frac{8 x ^{3}}{3} = 8 x^{3}$ .
Outer integral in $x$ . $I = \int_{0}^{4} 8 x^{3} d x = [2 x^{4}]_{0}^{4} = 2 \times 256 = 512$
Answer. The value is exactly $512$ , so rounding changes nothing: $I = 512$ .

KEY IDEA

Read the region off the figure and slice it the easy way. With vertical strips, the triangle bounded by $y = x$ , $y = - x$ and $x = 4$ becomes $0 \leq x \leq 4$ , $- x \leq y \leq x$ , and one iterated integral covers it. Horizontal strips would split it at $y = 0$ into two pieces. The integrand is even in $y$ and the region is symmetric about the x-axis, so the inner integral is twice the integral from $0$ to $x$ .

WATCH OUT FOR

Integrating over only the upper half, $0 \leq y \leq x$ , gives 256. So does integrating $x^{2} + y^{2}$ over the whole triangle and forgetting the factor 3 in front (that also gives 256, and 256/3 if only one of the two errors is made). Another slip is using $\int_{0}^{x} y^{2} d y = x^{2}$ in place of $x^{3} /3$ . Some students also try polar coordinates, where $x = 4$ becomes $r = 4 sec θ$ for $- π /4 \leq θ \leq π /4$ . That leads to a harder $sec^{4} θ$ integral with more chances to go wrong.

From the GATE 2022 EC paper. Every question of this paper · Download the full paper · All 39 Calculus questions, with notes

GATE 2022 EC Q.53: solution and answer

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