Integrals as accumulation

Definite Integrals as Accumulation

The definite integral measures signed area under a curve, built by adding infinite slices via Riemann sums:

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

Here, $[a,b]$ is partitioned into $n$ sub-intervals of width $\Delta x = \frac{b-a}{n}$ , with a sample point $x_i^*$ in each.

Property	Formula
Linearity	$\int_a^b [\alpha f + \beta g]\,dx = \alpha\int f + \beta\int g$
Additivity	$\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx$
Average Value	$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Common pitfall: A definite integral is a signed number, not pure area. Regions below the axis count negative; total area needs splits at zero crossings.

Core Rules & Applications

Integrals accumulate rates of change into total amounts. In physics, the Work done by a variable force is $W=\int_{a}^{b}F(x)\,dx$ .

To compute these, master basic antiderivatives:

Function	Antiderivative	Edge Case
Power rule $x^n$	$\frac{x^{n+1}}{n+1} + C$	$n \neq -1$
Reciprocal $1/x$	$\ln\|x\| + C$	Domain excludes $0$
Exponential $e^x$	$e^x + C$	Always true

The constant $C$ accounts for the family of all valid antiderivatives in indefinite integration.

Calculus: Integrals as accumulation

Definite Integrals as Accumulation

Core Rules & Applications

Practise this lesson

Calculus of a Single Variable