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GATE 2017 EE – Question 27

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 1 mark · Numerical answer

Let $I = c\iint_R xy^2\,dx\,dy$, where $R$ is the region shown in the figure and $c = 6 \times 10^{-4}$. The value of $I$ equals ________. (Give the answer up to two decimal places.)

The region $R$ is bounded by the $x$-axis below, the vertical lines $x = 1$ and $x = 5$ on the sides, and a straight line on top that runs from $(1, 2)$ to $(5, 10)$.

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Show answer and explanation

Correct answer: 0.99 to 1.01

Explanation

The slanted top edge passes through $(1, 2)$ and $(5, 10)$, so it is the line $y = 2x$. For each $x$ between 1 and 5, $y$ goes from 0 to $2x$. The inner integral is $\int_0^{2x} xy^2\,dy = x\frac{(2x)^3}{3} = \frac{8x^4}{3}$. Then $\int_1^5 \frac{8x^4}{3}\,dx = \frac{8}{15}(5^5 - 1) = 1666.13$. Multiplying by $c = 6 \times 10^{-4}$ gives $I = 1.00$.