Definite Integrals & Area Under a Curve

A definite integral attaches limits and produces a number:

$\int_a^b f(x) \,dx = [F(x)]_a^b = F(b) - F(a)$

No $+c$, it cancels in the subtraction. The layout is the working: bracketed antiderivative, then the substituted line written before evaluating, then the value. That middle line carries the method mark, especially on Paper 1 where the arithmetic lands in exact fractions.

$\int_1^3 (3x^2 - 2x) \,dx = [x^3 - x^2]_1^3 = (27 - 9) - (1 - 1) =$ 18

Area under a curve: sketch first, always

Area questions are definite integrals plus geometry awareness, and the awareness comes from a 30-second sketch:

1. Sketch the curve over the interval; mark axis crossings.
2. Identify the region, above the axis? Below? Both?
3. Set up the integral(s) accordingly.
4. Evaluate with the bracket layout.

Below-axis regions integrate negative. The integral isn’t wrong, it’s signed. For total area, split at the crossings and add magnitudes:

Curve crossing at $x = 2$ inside $[0, 5]$: Area $= \left|\int_0^2 f \,dx\right| + \left|\int_2^5 f \,dx\right|$ Integrating straight across $[0, 5]$ lets the signed parts cancel, the classic “lost area” error, and the sketch is the only reliable defence.

Area between a curve and a line (or two curves)

Area $= \int (\text{top} - \text{bottom}) \,dx$, between their intersection points

Find the intersections (they’re the limits), identify which function is on top across the interval (the sketch again), integrate the difference. Subtracting in the wrong order flags itself with a negative answer, fix the order, don’t just drop the sign silently. Composite regions (curve-area minus a triangle, say) decompose like circular-measure figures: name the pieces, compute, assemble.

Common mistakes

• The substituted $F(b) - F(a)$ line skipped, method mark forfeited
• Below-axis cancellation (no sketch, no split)
• Limits in the wrong order
• Between-curves integrals built from the wrong “top”
• Exact fractions abandoned for decimals mid-evaluation

Full topic context: Calculus notes · the full drill: integration & area technique.