Definite Integrals of Non-Negative Functions Examples

Identify the various parts of the expression

• the integrand

• the lower limit of integration

• the upper limit of integration

• the variable of integration

• The integrand is the part inside the integral symbol:

• The lower limit of integration is the number at the bottom of the integral sign:

• The upper limit of integration is the number at the top of the integral sign:

• The variable of integration is x:

Translate into symbols: "the integral of the function 2t + 3t² on the interval [0, 2]."

We know the answer will look something like this:

We need to fill in all the pieces. We're taking the integral of the function 2t + 3t², so that's our integrand:

Since this function uses the variable t, that's our variable of integration:

Since we're looking at the interval [0,  2], the lower limit of integration is 0 and the upper limit of integration is 2:

Let f (x) = 4x. Find .

It helps to look at a graph:

.

The area between f and the x-axis on this interval is a triangle with base 2 and height 8. We know the area of this triangle is

This means

Let f (x) = 4x + 2. Find .

From the graph, we can see that this area splits nicely into a rectangle and a triangle:

We get