Q.$f$ is continuous and an even function on $[-a, a]$. If $\displaystyle\int_{-a}^{a} f(x)\,dx = 2026$, then $\displaystyle\int_{-a}^{0} f(x)\,dx$ = ____.

The Even Function Property: A Mirror in Mathematics

Stand in front of a mirror: the distance from your nose to the mirror equals the distance from the mirror to your reflection. That's the core idea of an even function — it's symmetric about the vertical axis (the y-axis).

The Intuition

Take $f(x) = x^2$. At $x = 3$, $f(3) = 9$; at $x = -3$, $f(-3) = 9$ as well. The output is identical for a number and its negative — and this happens for every single $x$ in the domain.

Graphically, if you fold the paper along the y-axis, the left half of the graph lands exactly on top of the right half. The curve is a perfect mirror image of itself.

The Precise Statement

One equation — but it must hold for every $x$ where the function is defined, not just for a few nice numbers.

What This Means in Practice

If you know the value at $x = 5$, you automatically know the value at $x = -5$ — they're the same. This property lets you halve your work when analyzing the function.

Examples that satisfy the property:

• $f(x) = x^2$ (check: $(-x)^2 = x^2$)
• $f(x) = \cos x$ (check: $\cos(-x) = \cos x$)
• $f(x) = |x|$ (check: $|-x| = |x|$)
• $f(x) = x^4 - 3x^2 + 1$ (only even powers of $x$)

Why "Even"?

The name comes from even powers: $x^2$, $x^4$, $x^6$ all satisfy $(-x)^n = x^n$ when $n$ is even. Odd powers like $x^3$ give $(-x)^3 = -x^3$, which is a different property (odd functions).

A Quick Test

1. Replace every $x$ with $-x$ in the formula.
2. Simplify.
3. If you get back exactly the original expression, it's even. …