Definite integral

The exact signed area under a curve, $\int_a^b f(x)\,dx$ — the limit of a Riemann sum as $n\to\infty$ .

The definite integral $\int_a^b f(x)\,dx$ is the limit of a Riemann sum as the number of rectangles $n\to\infty$ (equivalently, as each rectangle's width $\Delta x\to0$ ): $\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$ . Unlike the indefinite integral, a definite integral is a single NUMBER — the exact signed area between the curve and the $x$ -axis, with area below the axis counted negatively.

Examples

$\int_0^2 x\,dx = 2$ (the area of a triangle with base $2$ and height $2$ ).
$\int_{-1}^{1} x^3\,dx = 0$ , since the positive and negative signed areas cancel exactly.