The value of $\frac{4}{\pi} \int_0^{\pi/2} \sin^2 x \text{ dx}$ is _________________ (rounded off to two decimal places).
Integral Evaluation: $\frac{4}{\pi} \int_0^{\pi/2} \sin^2 x \text{ dx}$
To find the value of the given expression, we need to evaluate the definite integral $\int_0^{\pi/2} \sin^2 x \text{ dx}$ and then multiply it by $\frac{4}{\pi}$.
Integral Step-by-Step
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Apply Trigonometric Identity: Use the identity $\sin^2 x = \frac{1 - \cos(2x)}{2}$.
The integral becomes:
$ \int_0^{\pi/2} \frac{1 - \cos(2x)}{2} \text{ dx} $
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Integrate the Expression: Find the antiderivative of the function.
$ \frac{1}{2} \int_0^{\pi/2} (1 - \cos(2x)) \text{ dx} = \frac{1}{2} \left[ x - \frac{\sin(2x)}{2} \right]_0^{\pi/2} $
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Apply Limits of Integration: Substitute the upper and lower limits.
$ \frac{1}{2} \left[ \left( \frac{\pi}{2} - \frac{\sin(\pi)}{2} \right) - \left( 0 - \frac{\sin(0)}{2} \right) \right] $
Since $\sin(\pi) = 0$ and $\sin(0) = 0$, this simplifies to:$ \frac{1}{2} \left[ \left( \frac{\pi}{2} - 0 \right) - (0 - 0) \right] = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4} $
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Calculate Final Value: Multiply the integral result by the constant factor $\frac{4}{\pi}$.
$ \text{Value} = \frac{4}{\pi} \times \frac{\pi}{4} $
$ \text{Value} = 1 $
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Rounding: Round the result to two decimal places.
The value is exactly 1. Rounded to two decimal places, it is 1.00.
This result aligns with the information that the correct value lies between 1 and 1, indicating the value is approximately 1.
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 ___________.- 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?
