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GATE 2022 ME (ME2) – Question 3

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

If $f(x)=2\ln(\sqrt{e^x})$, what is the area bounded by $f(x)$ for the interval $[0,2]$ on the $x$-axis?

  1. $\frac12$
  2. 1
  3. 2
  4. 4

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Correct answer: (C) 2

Explanation

**Step 1: simplify $f(x)$.**
$$f(x)=2\ln\!\big(\sqrt{e^{x}}\big)=2\ln e^{x/2}=2\cdot\frac{x}{2}=x.$$

**Step 2: area under $f(x)=x$ on $[0,2]$.** The function is non-negative there, so the area is the integral:
$$\int_0^2 x\,dx=\left[\frac{x^2}{2}\right]_0^2=2.$$

The area is **2**.