Properties of Definite Integrals

If $$\text{a} < \text{b}$$ , then $$\text{int}_{\text{a}}^{\text{b}} f(x)dx = -\text{int}_{\text{b}}^{\text{a}} f(x)dx$$ .

The value is 0 for any function f(x).

If $$\text{a} < \text{b} < \text{c}$$ , then $$\text{int}_{\text{a}}^{\text{c}} f(x)dx = \text{int}_{\text{a}}^{\text{b}} f(x)dx + \text{int}_{\text{b}}^{\text{c}} f(x)dx$$ .

If c is a constant, then $$\text{int}_{\text{a}}^{\text{b}} c \times f(x)dx = c \times \text{int}_{\text{a}}^{\text{b}} f(x)dx$$ .

The result is $$\text{int}_{-4}^{5} f(x)dx$$ .

The value is $$9 + (-11) = -2$$ .

The Fundamental Theorem of Calculus states that if F is an antiderivative of f, then $$\text{int}_{a}^{b} f(x)dx = F(b) - F(a)$$ .

The value is $$2 \times 14 = 28$$ .

The value is 0.

It means that the area under the curve f(x) from a to b is zero, which can occur if f(x) is positive and negative in equal measure.