# At a Glance - Definite Integrals of Real-Valued Functions

When we're integrating a non-negative function from a to b, the integral can be thought of as the "area under the curve" of the function. However, most of the time we can't count on having a non-negative function to integrate.

Assume f is a function that's allowed to take on negative values, and we're integrating from a to b with a < b. Then  is the weighted sum of the areas between the graph of f and the x-axis. We look at all areas between f and the x-axis. If they're on top of the x-axis we count them positively. If they're below the x-axis we count them negatively.

In other words, we add all the areas on top of the x-axis, then subtract all the areas below the x-axis.

#### Example 1

 Let f(x) = 2x. Find .

 Find

#### Example 3

 If , what is a?

#### Exercise 1

Find the integral.

where

#### Exercise 2

Find the integral.

#### Exercise 3

Find the integral.

#### Exercise 4

Find the integral.

#### Exercise 5

Find the integral.

where g(t) = 3t + 4.

#### Exercise 6

• If  and a is positive, what is a?

#### Exercise 7

• If  and a is positive, find a.

#### Exercise 8

• Find c given that .