Further Properties of the Definite Integral

The `area' interpretation of the definite integral gives us the following extra properties:

    if     f (x)$$\ge$$0    on     [a, b]    then     $$\int_{a}^{b}$$f (x) dx$$\ge$$ 0

(If the graph does not go below the x-axis the integral is not going to be negative.)

    if     f (x)$$\ge$$g(x)    on     [a, b]    then     $$\int_{a}^{b}$$f (x) dx$$\ge$$$$\int_{a}^{b}$$g(x) dx

(If the graph of f never goes below the graph of g then the area under the f graph is greater than that under the g graph. That explanation makes sense if both graphs are above the x-axis, but the result is true in general as you can see by using the first property on the function h(x) = f (x) - g(x).)

    if     m$$\le$$f (x)$$\le$$M    on     [a, b]    then     m(b - a)$$\le$$$$\int_{a}^{b}$$f (x) dx$$\le$$M(b - a)

(Think of m as the constant function g(x) = m and use the earlier results, together with $\int_{a}^{b}$m dx = m(b - a). Similarly for M.)

Finally, a property that is often quite useful

$$\left\vert\vphantom{\int_a^b f(x) dx}\right.$$$$\int_{a}^{b}$$f (x) dx$$\left.\vphantom{\int_a^b f(x) dx}\right\vert$$$$\le$$$$\int_{a}^{b}$$|f (x)| dx