We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Definite Integrals

Definite Integrals

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 .

Example 2


Example 3

If , what is a?

Exercise 1

Find the integral.


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 .

People who Shmooped this also Shmooped...