Definite Integrals: Definition

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  • 0:06 Definite Integrals
  • 1:26 Example #1
  • 2:31 Example #2
  • 3:52 Example #3
  • 6:16 Lesson Summary
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Lesson Transcript
Instructor: Erin Monagan

Erin has been writing and editing for several years and has a master's degree in fiction writing.

Explore how driving backwards takes you where you've already been as we define definite integrals. This lesson will also teach you the relationship between definite integrals and Riemann sums. Then, discover how an integral changes when it is above and below the x-axis.

Definite Integrals

Equation for the definite integral
Definite Integral Equation

Let's say you want to find the area between some line and the x-axis. You know that you could use a Riemann sum approach. This approach divides the region up into a bunch of different slices, and you estimate the area of each slice. As the width of each slice gets thinner, your estimate of the total area gets better. Eventually, you'll get the total area exactly. We write this as the limit as delta x goes to zero of the sum of all of the slices from k=1 to n of f(x sub k) * delta x sub k, or the height times the width of each slice.

This limit equals the definite integral from a to b of f(x)dx. This is the integral, but what does integral mean? We've said that it can mean the area under the curve, but does it always mean that?

Example #1

Graphing velocity as a function of time
Graphing Velocity as a Function of Time

Let's consider this example: You're driving and you graph your velocity as a function of time. Let's say your velocity is 20 mph, so let's graph that. You're driving 20 mph from t=0 to t=.5 hrs. If you're driving for a half an hour at 20 mph, how far have you gone? Well, 20 * .5 means that you've gone 10 miles, because the distance traveled is your velocity * time. On this particular graph, velocity is the height of the box and time is the width of the box, and velocity * time gives you the area of this box. In the case when you're graphing velocity as a function of time, the integral (the area underneath the curve) is also equal to the distance that you've traveled.

Example #2

Let's look at another case. Let's say that for the first half an hour, you're driving 20 mph because you're stuck in traffic. Then the next half hour you're going 40 mph; traffic has gotten a little bit lighter. Now how far have you traveled? The first half hour you traveled 10 miles. In the second half hour, you've traveled 20 miles, because 40 * .5 = 20). In total, you've traveled 10 + 20 miles, which is 30 miles. Again, this is the area underneath the curve of velocity as a function of time.

Area underneath the curve of velocity as a function of time
Velocity as a Function of Time Graph 2

In this case, the integral of velocity as a function of time gives you the area under the curve, which is your distance traveled. We can make this a little more specific and say that if your velocity is given as a function, f(t), and you're traveling from time a to time b, then the distance that you've traveled equals the integral from lower limit a to upper limit b of f(t)dt. In this case, you're integrating the function f(t) with respect to t.

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