Question

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?

• A. \( \frac{1}{2} \)
• B. 1
• C. 2
• D. 4

Hint:

When finding the area under a curve, simplify the function if possible and then integrate it over the given interval.

Answer 1

The Correct Option is C

We are asked to find the area bounded by the function \( f(x) = 2 \ln(\sqrt{e^x}) \) on the interval \([0, 2]\).
Step 1: Simplify the function.
We start by simplifying the given function: \[ f(x) = 2 \ln(\sqrt{e^x}) = 2 \ln(e^{x/2}) = x. \] Thus, \( f(x) = x \).
Step 2: Set up the integral.
The area under the curve \( f(x) \) from \( x = 0 \) to \( x = 2 \) is given by the definite integral: \[ \text{Area} = \int_{0}^{2} f(x) \, dx = \int_{0}^{2} x \, dx. \] Step 3: Evaluate the integral.
The integral of \( x \) is: \[ \int x \, dx = \frac{x^2}{2}. \] Evaluating from \( 0 \) to \( 2 \): \[ \left[ \frac{x^2}{2} \right]_{0}^{2} = \frac{2^2}{2} - \frac{0^2}{2} = \frac{4}{2} = 2. \] Step 4: Conclusion.
Thus, the area bounded by \( f(x) \) is 2.