Area Calculation for the Function \(f(x)\)

To determine the area bounded by a function \(f(x)\) and the x-axis over a specified interval, we typically use definite integration. Let's break down the process into simplifying the given function and then performing the integration.

Function Simplification

The provided function is \(f(x)=2\ln(\sqrt{e^x})\). It's always a good practice to simplify the function before proceeding with calculations like integration. We can use standard properties of exponents and logarithms to achieve this:

• First, let's simplify the term inside the logarithm: \(\sqrt{e^x}\). We know that a square root can be expressed as a power of \(\frac{1}{2}\). \[\sqrt{e^x} = (e^x)^{1/2}\]
• Next, apply the exponent rule \((a^m)^n = a^{mn}\): \[(e^x)^{1/2} = e^{x \cdot (1/2)} = e^{x/2}\]
• Now, substitute this simplified term back into the original function \(f(x)\): \[f(x) = 2\ln(e^{x/2})\]
• Finally, use the fundamental property of natural logarithms that \(\ln(e^k) = k\). In our case, \(k = x/2\): \[f(x) = 2 \cdot (x/2)\]
• Multiplying the terms, we get the remarkably simple function: \[f(x) = x\]

Thus, the problem is transformed into finding the area bounded by the function \(y=x\) (a straight line) on the x-axis for the interval [0, 2].

Area Bounded by \(f(x)\) on the X-axis

Since the simplified function is \(f(x)=x\) and the interval is [0, 2], the function is positive over this interval. Therefore, the area bounded by \(f(x)\) and the x-axis can be directly calculated by finding the definite integral of \(f(x)\) from 0 to 2.

The area \(A\) is given by the integral:

\[A = \int_{0}^{2} f(x) dx\]

Substituting \(f(x)=x\):

\[A = \int_{0}^{2} x dx\]

Now, we evaluate the definite integral:

• Find the antiderivative of \(x\). The power rule for integration states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). For \(x\) (which is \(x^1\)), the antiderivative is \(\frac{x^{1+1}}{1+1} = \frac{x^2}{2}\).
• Next, apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit (2) and subtracting its value at the lower limit (0): \[A = \left[ \frac{x^2}{2} \right]_{0}^{2}\] \[A = \left( \frac{(2)^2}{2} \right) - \left( \frac{(0)^2}{2} \right)\]
• Calculate the values: \[A = \left( \frac{4}{2} \right) - \left( \frac{0}{2} \right)\] \[A = 2 - 0\]
• The final result for the area is: \[A = 2\]

Conclusion for Area

The area bounded by the function \(f(x)=2\ln(\sqrt{e^x})\) for the interval [0, 2] on the x-axis is 2 square units. This calculation highlights the importance of simplifying complex functions before performing calculus operations.