How Do You Find the Area Under a Curve Using a Definite Integral?

You find the area under a curve by evaluating the definite integral of its function between the stated limits. If $f(x) \geq 0$ on ([a,b]), then the geometric area is $\displaystyle A=\int_{a}^{b} f(x)\,dx$.

The Reasoning

A definite integral accumulates infinitely thin vertical strips across an interval. Geometrically, it represents the signed area between $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$. Area above the axis is positive; area below it is negative.

This is IB Mathematics: Applications and Interpretation SL 5.5. At SL, examination questions on area under a curve use $f(x)>0$, so the integral directly equals the geometric area. A GDC is available on every Math AI paper.

For example, find the area under $y=x^2$ from $x=0$ to $x=3$:

$$A=\int_{0}^{3}x^2\,dx.$$

Using the power rule,

$$\int x^2\,dx=\frac{x^3}{3}+C.$$

Apply the upper and lower limits:

$$A=\left[\frac{x^3}{3}\right]_{0}^{3} =\frac{3^3}{3}-\frac{0^3}{3} =9.$$

Therefore, the area is $9$ square units. The constant $C$ is not included in the final evaluation because it cancels when the endpoint values are subtracted.

A common misconception is that every definite integral automatically gives geometric area. It gives signed area. If a curve crosses the axis, negative contributions can cancel positive ones; finding total geometric area would require splitting the interval at the roots and making each contribution positive. That extension is beyond the specific SL 5.5 case described here.

Exam Technique

1. State the boundaries and write the integral before evaluating it.
2. Use your GDC to evaluate the definite integral, but show the mathematical set-up rather than calculator keystrokes.
3. Keep full precision until the end, then give an exact answer or round to three significant figures unless instructed otherwise.
4. Include square units when the variables have physical units.

The command term calculate requires relevant working. The integral set-up can earn the method mark even if a later numerical error occurs.