GATE AE 2019 • Q24 NAT • 1 Mark Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Basic Fluid Mechanics

Subtopic: Conservation Equations of Mass, Momentum & Energy

Question Statement:

A function $f(x)$ is defined by $f(x) = \frac{1}{2}(x + |x|)$. The value of $\int_{-1}^{1} f(x)\,dx$ is _________ (round off to 1 decimal place).

Official Answer Key:

0.5

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2019 Q.24 / AE Q.14)** **1. Intuitive Physical Setup & Given Parameters**: - Function: $f(x) = \frac{1}{2}(x + |x|)$ - Integral limits: from $x = -1$ to $x = 1$. **2. Step-by-Step Mathematical Derivation**: 1. **Piecewise Definition of $|x|$**: - For $x < 0$: $|x| = -x \implies f(x) = \frac{1}{2}(x - x) = 0$ - For $x \ge 0$: $|x| = x \implies f(x) = \frac{1}{2}(x + x) = x$ (This is the standard Ramp function $R(x) = \max(0, x)$). 2. **Splitting the Integral at $x = 0$**: \int_{-1}^{1} f(x)\,dx = \int_{-1}^{0} 0\,dx + \int_{0}^{1} x\,dx 3. **Evaluating the Non-Zero Integral**: \int_{0}^{1} x\,dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1^2}{2} - 0 = \frac{1}{2} = 0.5 **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Analytical Evaluation**: **0.5** (Acceptable Range: [0.5, 0.5]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **0.5** - **Exam Rule**: Whenever $|x|$ appears in a definite integral, split the domain across $x = 0$.
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