Discussion
If \(f(x) = 2\ln\left(\sqrt{e^{x}}\right)\), what is the area bounded by \(f(x)\) for the interval [0, 2] on the \(x\)-axis?
Answer: C
The area is 2.
Simplify the function first. \(\sqrt{e^{x}} = e^{x/2}\), so
\[ \begin{aligned} f(x) &= 2\ln(e^{x/2}) = 2 \cdot \frac{x}{2} = x \\ \int_{0}^{2} x\, dx &= \frac{2^{2}}{2} = 2 \end{aligned} \]
The region is a right triangle with base 2 and height 2, area half of 4. Forgetting the factor 2 in front of the logarithm gives f(x) = x/2 and an area of 1; dropping the square root gives f(x) = 2x and an area of 4. 1/2 matches neither.
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