PYQ (JEE Main & Advanced)

High-Yield JEE Main Concepts for Application of Integrals

1. Standard Area Formulas

Memorizing standard formulas saves critical time during the exam.

• Area bounded by the parabola $y^2 = 4ax$ and the line $y = mx$ is $\frac{8a^2}{3m^3}$.
• Area bounded by two parabolas $y^2 = 4ax$ and $x^2 = 4by$ is $\frac{16ab}{3}$.
• Area of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $\pi ab$.

2. Symmetry and Even/Odd Functions

Always check for symmetry. If a region is symmetric about the y-axis, compute the integral from $0$ to $a$ and multiply by 2. If an integrand is odd over $[-a,a]$, the algebraic integral is zero, but the geometrical area is found using modulus or by doubling the positive part.

3. Shifting of Origin

If a parabola or circle is translated, shifting origin can simplify the equation without changing the area.

4. Modulus and Piecewise Functions

For $|f(x)|$, first find the roots of $f(x)=0$ in the interval, then split the integral and remove modulus according to sign.

5. Integration with respect to $y$

Whenever curves are naturally written as $x=f(y)$, it is usually better to integrate with respect to $dy$.