GATE BT 2024 • Q65 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Fundamentals of Calculus

Question Statement:

The absolute relative error in evaluating the integral ∫ x2dx 1 0 by the trapezoidal rule with the step size 0.25 is ________ % (rounded off to 2 decimal places).

Official Answer Key:

3.11 to 3.13

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2024 Q.65)** **1. Governing Concept & Definition**: - **Topic**: Numerical Integration & Error in Trapezoidal Rule - Comparing the exact analytical definite integral with the numerical approximation via the composite Trapezoidal Rule. **2. Detailed Step-by-Step Calculation**: - **Integral**: $\int_0^1 x^2 \, dx$ - **Exact Analytical Value**: $I_{exact} = \left[ rac{x^3}{3} ight]_0^1 = rac{1}{3} pprox 0.33333$ - **Trapezoidal Rule Approximation** (single strip with $h = 1$): $I_{trap} = rac{h}{2} [f(0) + f(1)] = rac{1}{2} [0 + 1] = 0.5$ *(For multiple intervals with step-size $h=0.25$, $I_{trap} pprox 0.34375$)* - **Relative Error**: $ ext{Relative Error} = rac{|I_{trap} - I_{exact}|}{I_{exact}} imes 100\% = 3.12\%$ **3. Final Answer**: - **Calculated Value**: **3.12%** - **Official GATE Accepted Range**: **3.11 to 3.13**

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