Interpret the definite integral as a limit of Riemann sums and as the signed area between a curve and the x-axis

What this dot point is asking

SCSA Unit 4 defines the definite integral as a limit of sums before connecting it to antidifferentiation. This dot point asks you to interpret the integral as accumulated signed area and to use its basic properties, supporting questions in both examination sections.

The limit of Riemann sums

Divide $[a,b]$ into $n$ thin strips of width $\Delta x$, and approximate the area under $y=f(x)$ by the sum of rectangle areas $\sum f(x_i)\Delta x$. As the strips become infinitely thin, this sum approaches the exact area.

This construction explains why the integral measures accumulated area: each rectangle contributes its height times its width, and the limit removes the approximation error.

Signed area

The height $f(x_i)$ is negative wherever the curve dips below the axis, so those strips contribute negative amounts. The definite integral therefore gives the signed area: positive above the axis, negative below.

Properties of the definite integral

These properties follow directly from the limit definition and are used to simplify calculations.

• $\displaystyle\int_a^a f(x)\,dx = 0$ (zero width).
• $\displaystyle\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$ (reversing the limits negates).
• $\displaystyle\int_a^c f(x)\,dx + \int_c^b f(x)\,dx = \int_a^b f(x)\,dx$ (additivity over adjacent intervals).
• $\displaystyle\int_a^b k\,f(x)\,dx = k\int_a^b f(x)\,dx$ (constants factor out).