Animated Solution for Mathematics - Definite Integration: The area enclosed within the curve $|x|+|y|=1$ is .........

The equation $|x|+|y|=1$ is a classic problem that tests your understanding of the modulus function and coordinate geometry. At first glance, it appears simple, but it serves as a masterclass in symmetry and piecewise analysis.

The modulus function, $|a|$, acts as a gatekeeper of sign. It dictates that if $a$ is positive, it remains unchanged, but if $a$ is negative, its sign is flipped. Applying this to both $x$ and $y$ creates a piecewise function that changes definition based on the quadrant of the Cartesian plane.

Imagine standing at the origin $(0,0)$. As you move into the first quadrant, where both $x\ge 0$ and $y\ge 0$, the equation $|x|+|y|=1$ simplifies to:

This represents a straight line with intercepts at $(1,0)$ and $(0,1)$.

Now, pivot to the second quadrant where $x<0$ and $y\ge 0$. Here, $|x|$ becomes $-x$, transforming the equation into:

This line connects $(0,1)$ to $(-1,0)$. As you rotate through the third and fourth quadrants, the signs continue to flip, creating four distinct line segments that enclose a geometric region.

Many students might attempt to use integration to find this area, but that is unnecessary given the inherent symmetry of the shape. The equation $|x|+|y|=1$ is symmetric about both the $x$-axis and the $y$-axis.

The resulting shape is a diamond, or a square tilted at $45^{\circ }$. Because of this symmetry, the area in the first quadrant is identical to the area in the other three quadrants.

We can simply calculate the area of the triangle in the first quadrant and multiply it by $4$. The triangle in the first quadrant has vertices at $(0,0)$, $(1,0)$, and $(0,1)$.

This is a right-angled triangle with a base of $1$ unit and a height of $1$ unit. Using the standard area formula:

Substituting our values:

Since there are four such identical triangles forming our diamond, the total area is:

This approach is elegant and efficient, saving precious time during the JEE Advanced exam.

Remember this general rule: for any equation of the form $|x|+|y|=a$, the enclosed area is always $2a^2$. Keep this in your mental toolkit to master the modulus function with confidence.