GATE 2020 EC Q.51: solution and answer

GATE 2020 · ECQ.51 · 2 marks · NAT

For the solid $S$ shown below, the value of $∭_{S} x d x d y d z$ (rounded off to two decimal places) is ________.

GATE 2020 EC question 51: A 3-D sketch with the z-axis pointing up, the y-axis pointing right and the x-axis coming toward the viewer at the lower… — Figure from the GATE 2020 EC paper. It is part of the question.

Answer and solution

ANSWER

$∭_{S} x d x d y d z = \frac{9}{4} = 2.25$

Official answer: 2.25Source: Official answer key, GATE 2020 EC (IIT Delhi)

Read the region off the figure. At every fixed $x$ the cross-section is the same right triangle in the $y z$ -plane, with vertices $(y, z) = (0, 0)$ , $(1, 0)$ and $(0, 1)$ , and the prism runs from $x = 0$ to $x = 3$ . So $S = {0 \leq x \leq 3, y \geq 0, z \geq 0, y + z \leq 1}$ .
Set up the iterated integral with limits $0 \leq z \leq 1 - y$ , $0 \leq y \leq 1$ and $0 \leq x \leq 3$ : $I = \int_{0}^{3} \int_{0}^{1} \int_{0}^{1 - y} x d z d y d x$
The integrand depends only on $x$ , and the $y$ and $z$ limits do not depend on $x$ , so the integral separates (Fubini): $I = (\int_{0}^{3} x d x) (\int_{0}^{1} \int_{0}^{1 - y} d z d y)$
The $y$ – $z$ part is the area of the triangular cross-section: $\int_{0}^{1} (1 - y) d y = [y - \frac{y ^{2}}{2}]_{0}^{1} = \frac{1}{2}$ .
The $x$ part: $\int_{0}^{3} x d x = \frac{x ^{2}}{2}_{0}^{3} = \frac{9}{2}$ .
Multiply the two: $I = \frac{9}{2} \times \frac{1}{2} = \frac{9}{4} = 2.25$ .
Check: $I = \overset{x}{ˉ} V$ . The volume is $V = \frac{1}{2} \times 3 = \frac{3}{2}$ , and because every cross-section is the same, the centroid sits halfway along the length, $\overset{x}{ˉ} = \frac{3}{2}$ . That gives $\frac{3}{2} \cdot \frac{3}{2} = 2.25$ .

KEY IDEA

The solid is a right prism: its cross-section at every $x$ is the same triangle, of area $\frac{1}{2}$ . So the triple integral of $x$ splits into (area of the cross-section) $\times \int_{0}^{3} x d x$ . Put another way, it equals volume $\times$ the $x$ -coordinate of the centroid: $\frac{3}{2} \times \frac{3}{2}$ .

WATCH OUT FOR

Misreading the 3-D sketch. Some students take the triangle to lie in the $x y$ - or $x z$ -plane with the length 3 along $y$ or $z$ , which changes the integrand's weighting and gives the wrong value. Others treat the solid as a full box $[0, 3] \times [0, 1] \times [0, 1]$ and get $4.5$ , because they miss the slanted face $y + z = 1$ . A third mistake is to compute only the volume, $1.5$ , and forget the factor $x$ in the integrand.

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GATE 2020 EC Q.51: solution and answer

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