Module 5: Section 4, Part 2
The Definite Integral
In the last section we were introduced to the definite integral and the fundamental theorem of calculus which gives us a way to evaluate the definite integral. In this section we will discuss the concept of the definite integral more in depth, look at some applications and theorems involving the definite integral, and apply our antiderivative techniques to evaluate definite integrals.
First, let's recall what we get from the definite integral:
Click here if you would like to see a short video of a graphical representation of the definition of the definite integral.
The Fundamental Theorem of Calculus
Ok, now that we have those reminders, let's evaluate a few definite integrals.
Recall that the fundamental theorem of calculus says that we can calculate a definite integral if we can find the antiderivative of the integrand.
Let's take the following example step by step.
Try to calculate the following definite integrals, and then few the LiveScribe videos to check your answers:
LiveScribe Solution PDF Version
In this video the instructor calculates a basic definite integral. Follow this link.
In this video the instructor calculates a definite integral using u-substitution. Follow this link.
Now if we have definite integrals, we also have indefinite integrals (or general antiderivatives). Let's take a look at how the characteristics of the two integrals differ:
We took a quick look at two theorems in the last section. Let's revisit those, but this time for definite integrals.
The following video discusses why these two theorems are true from a grpahical standpoint.
If you would like to see proofs of some of these properties, follow this link.
Use the following two definite integrals to answer the questions
We saw in the last section that if we integrate a velocity function over an interval of time, we get the distance traveled over that interval. Let's look at a couple other scenarios.
Listen to the description here: