Animated Solution for Mathematics - Definite Integration: The value of ${\int }_{\pi }^{2\pi }[2\sin x]dx$ where $[.]$ represents the greatest integer function is

The integral of a Greatest Integer Function, such as $I={\int }_{\pi }^{2\pi }[2\sin x]dx$, is often perceived as a complex challenge. However, by viewing the Greatest Integer Function $[f(x)]$ as a "staircase" that remains constant between specific jump points, we can simplify the problem into manageable geometric segments.

Our goal is to identify the exact points where the value of $2\sin x$ transitions between integers.

We are integrating over the interval $[\pi ,2\pi ]$, which corresponds to the third and fourth quadrants. In this domain, the sine function is non-positive, ranging from $-1$ to $0$.

Consequently, the expression $2\sin x$ ranges from $-2$ to $0$. We must determine the specific values of $x$ where $2\sin x$ crosses the integers $-1$ and $-2$.

To find the jump points, we solve for the boundaries:

1. Setting $2\sin x=-1$ yields $\sin x=-1/2$. In the interval $[\pi ,2\pi ]$, this occurs at $x=7\pi /6$ and $x=11\pi /6$.

2. Setting $2\sin x=-2$ yields $\sin x=-1$, which occurs at $x=3\pi /2$.

These critical points divide our domain into three distinct regions where the function $[2\sin x]$ remains constant.

We now partition the integral based on these regions:

For $x\in [\pi ,7\pi /6)$, $2\sin x\in (-1,0)$, so $[2\sin x]=-1$. For $x\in [7\pi /6,11\pi /6)$, $2\sin x\in [-2,-1)$, so $[2\sin x]=-2$. * For $x\in [11\pi /6,2\pi ]$, $2\sin x\in (-1,0)$, so $[2\sin x]=-1$.

The integral $I$ is expressed as the sum of these three areas:

We calculate the area of each rectangular slice individually:

1. The first integral: $-1\times (7\pi /6-\pi )=-\pi /6$.

2. The second integral: $-2\times (11\pi /6-7\pi /6)=-2\times (4\pi /6)=-4\pi /3$.

3. The third integral: $-1\times (2\pi -11\pi /6)=-\pi /6$.

Summing these values, we obtain:

The final result is $-5\pi /3$.