GATE 2021 AE – Question 48
Approximate $\int_1^5x^2\,dx$ using four equal intervals. The difference between trapezoidal and Simpson results is (two decimals)
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Correct answer: 0.66 to 0.68
Explanation
$\int_1^5x^2\,dx$ with four equal intervals gives $h=\dfrac{5-1}{4}=1$ and nodes $x=1,2,3,4,5$ with $f=x^2=1,4,9,16,25$.
**Trapezoidal rule:**
$$T=\frac h2\big[f_0+2(f_1+f_2+f_3)+f_4\big]=\frac12\big[1+2(4+9+16)+25\big]=\frac12(84)=42 .$$
**Simpson's 1/3 rule:**
$$S=\frac h3\big[f_0+4(f_1+f_3)+2f_2+f_4\big]=\frac13\big[1+4(4+16)+2(9)+25\big]=\frac13(124)=41.333 .$$
Simpson's rule is exact for polynomials up to degree 3, and the exact value is $\dfrac{125-1}{3}=41.333$.
**Difference:**
$$T-S=42-41.333=0.667\approx\mathbf{0.67}.$$