Meaning of definite integrals as area, foundational idea of finding area using integration.
Class 12 · Mathematics · CBSE & JEEIn Chapter 7 we studied how to compute integrals. In Chapter 8, we use those integrals to find geometric areas — the actual area enclosed by curves, axes, and lines on the coordinate plane.
The central idea is: a definite integral \(\int_a^b f(x)\,dx\) is not just a number — it is the signed area between the curve \(y = f(x)\) and the \(x\)-axis over the interval \([a, b]\).
Every definite integral has a direct geometric meaning — the area under/above a curve segment.
We apply the tools of integration to measure real 2-D regions, which has uses in physics, economics, and engineering.
Standard curves: parabola, circle, ellipse, straight lines. Finding area bounded by them is directly asked in board exams.
Imagine slicing the region under a curve into infinitely thin vertical rectangles of width \(dx\). Each strip has height \(f(x)\) and area \(f(x)\,dx\). Adding all strips from \(x=a\) to \(x=b\) gives the total area.
Here \(f(x) \geq 0\) on \([a,b]\). The area is measured between the curve and the \(x\)-axis.
Write the curve — identify \(y = f(x)\).
Find limits — determine \(x = a\) and \(x = b\) (intersections with axis or given lines).
Integrate — evaluate \(\int_a^b f(x)\,dx\).
Take absolute value if curve goes below the \(x\)-axis in part of the interval.
The definite integral is the limit of the Riemann sum: \(\displaystyle A = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x\). As \(\Delta x \to 0\), the sum converges to the exact area.
Depending on the curve, it is sometimes easier to integrate with respect to \(y\) (horizontal strips) instead of \(x\).
| Strip Type | Formula | When to Use |
|---|---|---|
| Vertical (dx) | \(\displaystyle\int_a^b f(x)\,dx\) | Curve expressed as \(y = f(x)\); bounded top–bottom |
| Horizontal (dy) | \(\displaystyle\int_c^d g(y)\,dy\) | Curve expressed as \(x = g(y)\); bounded left–right |
| Curve | Equation | Key Feature |
|---|---|---|
| Parabola | \(y^2 = 4ax\) | Opens right; vertex at origin |
| Parabola | \(x^2 = 4ay\) | Opens upward; vertex at origin |
| Circle | \(x^2 + y^2 = r^2\) | Centre origin, radius r |
| Ellipse | \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) | Semi-axes a, b |
| Line | \(y = mx + c\) | Used as boundary |