Summary and Quick Revision

The Chapter in One Sweep

The definite integral is an area. Everything in this chapter follows from slicing a region into thin strips and summing them.

The two area formulas

1. Vertical strips: region bounded by $y = f(x)$, the $x$-axis, $x = a$ and $x = b$: $$A = \int_a^b y\,dx = \int_a^b f(x)\,dx$$
2. Horizontal strips: region bounded by $x = g(y)$, the $y$-axis, $y = c$ and $y = d$: $$A = \int_c^d x\,dy = \int_c^d g(y)\,dy$$

The sign rule

Below the axis, the integral is negative; report $|A_1| + A_2$, splitting at every axis crossing.

The showcase results

1. Circle $x^2 + y^2 = a^2$: area $= \pi a^2$, via $4\int_0^a\sqrt{a^2 - x^2}\,dx$.
2. Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: area $= \pi ab$ ($a, b$ = square roots of the denominators).
3. Building blocks: $\int_0^a\sqrt{a^2 - x^2}\,dx = \frac{\pi a^2}{4}$ (quarter circle); one arch of $\sin$ or $\cos$ has area $2$; a parabolic arch fills $\frac23$ of its bounding rectangle; the latus-rectum region of $y^2 = 4ax$ has area $\frac{8a^2}{3}$.

The JEE add-on: area between curves

$$A = \int_a^b\left[f(x) - g(x)\right]dx \quad (f \text{ on top; } a, b \text{ from } f = g)$$

Remember the classics: $y = x$ vs $y = x^2$ gives $\frac16$; $y^2 = 4ax$ vs $x^2 = 4by$ gives $\frac{16ab}{3}$.

The Decision Checklist and Mistake List

Decision checklist — run it on every problem

1. Sketch the region (ten seconds, sign-correct).
2. Boundaries: axis + ordinates? Two curves? A modulus corner?
3. Intercepts/intersections inside the interval → split points.
4. Symmetry: both-axes (circle/ellipse: $\times4$), even ($\times2$), point symmetry (congruent lobes).
5. Strip direction: does the curve solve cleanly for $y$ (use $dx$) or for $x$ (use $dy$)?
6. Integrate, take absolute values piecewise, add, attach "square units".

The mistake checklist — read before every exam

1. Integrating across an axis crossing without splitting — the chapter's number-one error; cancellation corrupts the area silently.
2. Reporting a negative area — the integral's sign is information (below the axis / wrong curve on top), never the final answer.
3. $\left|\int f\right|$ instead of $\int|f|$ — the absolute value goes on each piece, not on the total.
4. Ellipse area from the denominators — $\pi ab$ needs the square roots: $\frac{x^2}{16} + \frac{y^2}{9} = 1$ gives $12\pi$, never $144\pi$.
5. Forgetting $\times 4$ (or $\times 2$) after computing one quadrant of a symmetric figure.
6. Chain-rule amnesia in the anti-derivative: $\int\sin 2x\,dx = -\frac{\cos 2x}{2}$ — each arch of $\sin 2x$ has area $1$, not $2$.
7. Latus-rectum and chord regions: the region is symmetric about the parabola's axis — set up one half and double.
8. Wrong top curve between two curves — test one interior point before integrating $f - g$.
9. Fake intersections from squaring while solving $f = g$ — verify each root on the original equations.
10. Mixing the region's description: "bounded by the $y$-axis" means horizontal strips and $y$-limits — don't autopilot into $\int y\,dx$.

Key Point: Revision strategy: one pass through the formula blocks above, one pass through this list, then the Section 7 drill under time. Every wrong drill answer maps to exactly one checklist line — fix the line, not just the question.

[JEE Tip] The night before the exam, re-derive the circle's $\pi a^2$ once by strips and re-solve $y = x$ vs $y = x^2$ — one derivation and one between-curves classic lock the whole chapter's machinery into place.