Function f is known at the following points: The value of 03 f(x) dx computed using the trapezoidal rule is A) 8.983 B) 9.003 C) 9.017 D) 9.045

Official answer: D) 9.045. Explanation: From the given table, step size h = 0.3, x0 = 0, xn = 3.0 with n = 10 subintervals. The Trapezoidal Rule formula is: x0xn f(x) dx (h)/(2) [ f(x0) + f(xn) + 2Σi=1n-1 f(xi) ] Here: - f(x0) = f(0) = 0 - f(xn) = f(3.0) = 9.00 - Sum of intermediate terms: Σi=19 f(xi) = 0.09 + 0.36 + 0.81 + 1.44 + 2.25 + 3.24 + 4.41 + 5.76 + 7.29 = 25.65 Applying the formula: 03 f(x) dx (0.3)/(2) [ 0 + 9.00 + 2(25.65) ] 03 f(x) dx 0.15 × [9.00 + 51.30] = 0.15 × 60.30 = 9.045 Hence, the correct option is (D).

Calculus

GATE CSE 2013 · Q23

MCQ1 markMedium

Time00:00

Function $f$ is known at the following points: The value of $\int_0^3 f(x)\,dx$ computed using the trapezoidal rule is

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Solution

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Official answer: D) 9.045

Calculus

From the given table, step size $h = 0.3$ , $x_0 = 0, x_n = 3.0$ with $n = 10$ subintervals.

The Trapezoidal Rule formula is:

\int_{x_0}^{x_n} f(x)\,dx \approx \frac{h}{2} \left[ f(x_0) + f(x_n) + 2\sum_{i=1}^{n-1} f(x_i) \right]

Here:

$f(x_0) = f(0) = 0$
$f(x_n) = f(3.0) = 9.00$
Sum of intermediate terms:
$\sum_{i=1}^{9} f(x_i) = 0.09 + 0.36 + 0.81 + 1.44 + 2.25 + 3.24 + 4.41 + 5.76 + 7.29 = 25.65$

Applying the formula:

\int_0^3 f(x)\,dx \approx \frac{0.3}{2} \left[ 0 + 9.00 + 2(25.65) \right]

\int_0^3 f(x)\,dx \approx 0.15 \times [9.00 + 51.30] = 0.15 \times 60.30 = 9.045

Hence, the correct option is (D).

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This Calculus previous year question is Q23 of the GATE CSE 2013 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

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