Integrals as accumulation

The definite integral measures the signed area under a curve:

$$\int_{a}^{b} f(x)\,dx \;=\; \lim_{n\to\infty}\sum_{i=1}^{n} f(x_{i}^{*})\,\Delta x$$

Riemann sums — Partition $[a,b]$ into $n$ sub-intervals of width $\Delta x = \tfrac{b-a}{n}$, pick a sample point $x_i^*$ in each, and sum.

Properties

• Linearity:

$$\int_a^b [\alpha f + \beta g]\,dx = \alpha\int_a^b f\,dx + \beta\int_a^b g\,dx$$

• Additivity:

$$\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx$$

• Average value:

$$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$$

Basic antiderivatives

• $x^{n}$ → $\frac{x^{n+1}}{n+1}+C$ (for $n\neq -1$)
• $1/x$ → $\ln|x|+C$
• $e^{x}$ → $e^{x}+C$

Physics link: Work done by a variable force is $W=\int_{a}^{b}F(x)\,dx$.

Common pitfall: A definite integral is a signed number, not an area: regions below the axis count negative. "Total area enclosed" questions require splitting at the zero crossings first.