JEE Maths — Application of Integrals Complete Chapter Guide

Question

How do we use definite integrals to find the area bounded by curves? Walk through the method for area between two curves, area under a parabola, and choosing between horizontal and vertical strips.

(JEE Main — method selection + computation)

Solution — Step by Step

The area between a curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$ :

A = \int_a^b |f(x)|\,dx

The absolute value is critical — if the curve dips below the x-axis, integrate the positive and negative parts separately.

For two curves $y = f(x)$ (upper) and $y = g(x)$ (lower):

A = \int_a^b [f(x) - g(x)]\,dx

Example: Area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$ .

Here $y = x$ is above $y = x^2$ in $[0, 1]$ :

A = \int_0^1 (x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \mathbf{\frac{1}{6}}

If the curves are easier to express as $x = f(y)$ , integrate with respect to $y$ :

A = \int_{y_1}^{y_2} [x_{\text{right}} - x_{\text{left}}]\,dy

Use horizontal strips when: the bounding curves are of the form $x = (\text{something in } y)$ , or when vertical strips would require splitting the integral into multiple parts.

Problem Type	Approach
Area under parabola $y = ax^2$	Vertical strips, straightforward
Area of ellipse	$A = \pi ab$ (derive via $\int$ or use formula)
Area between parabola and line	Find intersection points first, then integrate difference
Area bounded by $	x

Find the area enclosed between $y^2 = 4x$ and $y = x$ .

Intersection: $y^2 = 4y \Rightarrow y = 0, 4$ , so $x = 0, 4$ .

Using horizontal strips (easier here since parabola gives $x = y^2/4$ ):

A = \int_0^4 \left(y - \frac{y^2}{4}\right)\,dy = \left[\frac{y^2}{2} - \frac{y^3}{12}\right]_0^4 = 8 - \frac{16}{3} = \mathbf{\frac{8}{3}}

graph TD
    A["Area Problem"] --> B{"Curves given as y = f of x?"}
    B -->|Yes| C["Vertical strips: integrate dx"]
    B -->|No| D{"Curves as x = g of y?"}
    D -->|Yes| E["Horizontal strips: integrate dy"]
    C --> F["Find intersection points"]
    E --> F
    F --> G["Identify upper/right curve"]
    G --> H["Integrate: upper minus lower"]
    H --> I["Check: use symmetry if possible"]
    style A fill:#fbbf24,stroke:#000,stroke-width:2px
    style H fill:#86efac,stroke:#000

Why This Works

The definite integral geometrically represents the signed area between a curve and an axis. When we compute $\int_a^b [f(x) - g(x)]\,dx$ , we are summing up infinitesimally thin vertical rectangles, each of height $f(x) - g(x)$ and width $dx$ . The choice between vertical and horizontal strips is purely about convenience — both give the same answer.

The skill tested in JEE is not the integration itself (which is usually simple) but the setup: finding intersection points correctly, identifying which curve is on top, and choosing the right variable of integration.

Common Mistake

The number one error: forgetting to find intersection points or finding them incorrectly. If your limits of integration are wrong, the entire answer is wrong — even if the integration is perfect. Always solve $f(x) = g(x)$ carefully and verify by plugging back. Also, do not forget the absolute value when a curve crosses the axis within the integration interval.

For JEE Main, about 1 question per paper comes from this chapter. Most are straightforward area-between-curves problems. The parabola-line and parabola-parabola combinations are the most common. Practise these 10 standard types and you will cover 90% of what appears.

Question

Solution — Step by Step

Why This Works

Common Mistake

Try These Next