Properties of Definite Integrals

If \( a < b < c \), then \( \int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx \).

You can use the property of additivity: \( \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx \).

It indicates that the area under the curve of the function \( f(x) \) from \( a \) to \( b \) is below the x-axis, resulting in a net negative area.

The definite integral \( \int_a^b f(x)dx \) represents the net area between the curve \( f(x) \) and the x-axis from \( x=a \) to \( x=b \).

If \( f(x) \) is odd, then \( \int_{-a}^a f(x)dx = 0 \).

Reversing the limits of integration changes the sign: \( \int_a^b f(x)dx = -\int_b^a f(x)dx \).

You can use the property of additivity to combine the values of the integrals over the subintervals.

The Fundamental Theorem of Calculus states that if \( F \) is an antiderivative of \( f \), then \( \int_a^b f(x)dx = F(b) - F(a) \).

The area above the x-axis contributes positively to the integral, while the area below the x-axis contributes negatively.

Evaluate the integral over each piece separately and then sum the results.