The value obtained by integrating a function over a specific interval [a, b], representing the area under the curve.

The value 'b' in the definite integral $\int_{a}^{b} f(x) , dx$, representing the upper bound of the interval.

The value 'a' in the definite integral $\int_{a}^{b} f(x) , dx$, representing the lower bound of the interval.

The function $f(x)$ being integrated in a definite integral.

$\int_{a}^{a} f(x) , dx = 0$

$\int_{b}^{a} f(x) , dx = -\int_{a}^{b} f(x) , dx$

$\int_{a}^{b} k cdot f(x) , dx = k \int_{a}^{b} f(x) , dx$

$\int_{a}^{b} [f(x) pm g(x)] , dx = \int_{a}^{b} f(x) , dx pm \int_{a}^{b} g(x) , dx$

$\int_{a}^{b} f(x) , dx + \int_{b}^{c} f(x) , dx = \int_{a}^{c} f(x) , dx$

The integral represents the area under the curve. If the upper and lower limits are the same, the 'width' of the area is zero, resulting in zero area.

Reversing the limits changes the direction of integration, thus negating the value of the integral because the area is now considered 'negative'.

Multiplying a function by a constant scales its area by the same factor. Thus, the constant can be factored out.

The total area under a curve from a to c can be found by summing the area from a to b and then from b to c.