Animated Solution for Mathematics - Definite Integration: ${\int }_0^{10\pi }|\sin x|dx$ is

Imagine you are standing on the $x$-axis, looking at the graph of $f(x)=|\sin x|$. Usually, the sine function is a wave that dances above and below the axis, a perfect balance of positive and negative.

But the modulus operator—those two vertical bars—acts like a mirror. It takes every negative 'valley' and flips it upwards, transforming the sine wave into a continuous, rhythmic series of identical humps.

This is the geometric reality of our problem: we are calculating the total area under these humps from $x=0$ to $x=10\pi$.

To solve this, we must first identify the 'heartbeat' of the function. A standard sine wave repeats every $2\pi$.

But because we have reflected the negative parts, the pattern repeats much faster. Look closely at the graph: the function returns to its starting shape after every $\pi$ interval. Therefore, the fundamental period $T$ of $|\sin x|$ is $\pi$.

In the world of JEE Advanced, recognizing periodicity is often the key to unlocking a problem. We use the property:

Here, our upper limit is $10\pi$, and our period $T$ is $\pi$. This means we have exactly $n=10$ identical humps. Instead of wrestling with a complex integral over a large range, we can simply find the area of one single hump and multiply it by $10$.

Now, let us focus on just one hump, the integral from $0$ to $\pi$. In this interval, $\sin x$ is non-negative, so the modulus is redundant: $|\sin x|=\sin x$.

Our integral simplifies to:

The anti-derivative of $\sin x$ is $-\cos x$. Applying the Fundamental Theorem of Calculus:

To avoid any sign errors, let us pull the negative sign outside:

Substituting the limits, we get:

Knowing that $\cos \pi =-1$ and $\cos 0=1$, the expression becomes:

The negative signs cancel out, leaving us with a clean, positive result of 20.

We started with a daunting integral, but by visualizing the geometry and applying the property of periodicity, we reduced it to a simple arithmetic task.

The area of a single hump of $|\sin x|$ is exactly $2$ square units. With $10$ such humps, the total area is 20.

Always remember: when you see modulus signs in a trigonometric integral, stop and look for the symmetry. It is often the shortest path to the answer.