Simplifying the Function $f(x)$
First, simplify the given function $f(x) = 2 \ln(\sqrt{e^x})$.
- Use the property $\sqrt{a} = a^{1/2}$: $f(x) = 2 \ln((e^x)^{1/2})$
- Use the exponent rule $(a^m)^n = a^{mn}$: $f(x) = 2 \ln(e^{x/2})$
- Use the logarithm property $\ln(e^a) = a$: $f(x) = 2 \times (x/2)$
- Simplify the expression: $f(x) = x$
The function simplifies to $f(x) = x$.
Calculating the Bounded Area
The area bounded by the function $f(x) = x$ on the interval $[0, 2]$ on the x-axis is calculated using a definite integral.
- Set up the integral for the area: Area = $\int_{0}^{2} f(x) dx$
- Substitute the simplified function $f(x) = x$: Area = $\int_{0}^{2} x dx$
- Find the antiderivative of $x$: The antiderivative is $\frac{x^2}{2}$.
- Evaluate the definite integral using the fundamental theorem of calculus: Area = $[\frac{x^2}{2}]_{0}^{2}$
- Substitute the upper and lower limits: Area = $\frac{2^2}{2} - \frac{0^2}{2}$
- Calculate the result: Area = $\frac{4}{2} - 0 = 2$
The area bounded by the function $f(x) = 2 \ln(\sqrt{e^x})$ over the interval $[0, 2]$ is 2.
Important Questions from Area Under Curve
- A function $y(x)$ is defined in the interval $[0, 1]$ on the x-axis as
$y(x) = \begin{cases} 2 & \text{if } 0 \le x < \frac{1}{3} \\ 3 & \text{if } \frac{1}{3} \le x < \frac{3}{4} \\ 1 & \text{if } \frac{3}{4} \le x \le 1 \end{cases}$
Which one of the following is the area under the curve for the interval $[0, 1]$ on the x-axis? - The area of the region bounded by the parabola $x = -y^2$ and the line $y = x + 2$ equals
The work done by the force $F = (x + y)\hat{i} - (x^2 + y^2)\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in $\vec{OX}$ and $\vec{OY}$ directions, respectively, along the upper half of the circle $x^2 + y^2 = 1$ from $(1,0)$ to $(-1,0)$ in the $xy$-plane is
In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.
The probability that any point picked randomly within the square falls in the shaded area is ___________.- Define $[x]$ as the greatest integer less than or equal to $x$, for each $x \in (-\infty,\infty)$. If $y = [x]$, then area under $y$ for $x \in [1,4]$ is
