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. Since \(\sqrt{e^{x}} = e^{x/2}\),
\[ \begin{aligned} f(x) &= 2\ln\left(e^{x/2}\right) \\ &= 2 \cdot \frac{x}{2} = x \end{aligned} \]
So the region is the triangle under the line \(y = x\) between \(x = 0\) and \(x = 2\):
\[ \int_{0}^{2} x\,dx = \frac{x^{2}}{2}\Big|_{0}^{2} = 2 \]
Dropping the factor 2 in front of the logarithm gives \(f(x) = x/2\) and an area of 1, and halving once more gives 1/2. 4 is \(x^{2}\) at \(x = 2\), the antiderivative without its division by 2.
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