Linear Properties of Definite Integrals

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  • 1:06 Zero Integral Property
  • 1:56 Backward Property
  • 2:49 Constant Property
  • 4:00 Additive Property
  • 4:57 Sums Property
  • 6:12 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.

If you're having integration problems, this lesson will relate integrals to everyday driving examples. We'll review a few linear properties of definite integrals while practicing with some problems.

The Integral

Remember that an integral is defined between a lower limit (x=a) and an upper limit (x=b) and you're integrating over f(x), which is known as the integrand. The variable of integration is written in this dx term, so in this case, we're integrating over x. We often think of this as being the area under a curve. Here, it's the area between f(x) and the x-axis (between x=a and x=b). Let's think of some of the properties these integrals have. For the sake of all these examples, let's actually integrate the function of your velocity, so velocity as a function of time. We have the integral from a to b of f(t)dt. Time is our independent variable.

The Backward Property of definite integrals
Backward Property

Zero Integral Property

The first property is the 'Going Nowhere Property'. This is really the Zero Integral Property. Let's say you've got f(t) (your velocity as a function of time) and you want to integrate from t=a to t=a f(t)dt. Remember that if you take the integral of your velocity as a function of time, it will give you how far you've gone over that period of time. In this case, time goes from a to a, so no time has elapsed. If no time has elapsed, you have gone nowhere. So the integral from a to a of f(t)dt=0, because no time has elapsed and you have gone nowhere.

Backward Property

The Constant Property
Constant Property

What if I write f(t)dt from t=a to t=b? This is going forward in time from t=a to t=b. What if I take the integral from t=b to t=a? This is going backward in time. If I go 30 miles forward from time a to time b, when I reverse time, I'm going to go 30 miles backward. So in terms of integrals, we write this as the integral from a to b of f(t)dt = - the integral from b to a of f(t)dt. This is the Backward Property. If you swap the limits of integration, here, you also have to swap the sign.

Constant Property

This Constant Property is also called the 'Speed Up Property' or the 'Do It Again Property'. Let's say you have an integral from a to b of C * f(t)dt, with C as a constant. Think of this as if Cf(t)=60 mph. If you go 60 mph from time a to time b, you're going to end up somewhere down the road. If instead, you are going 30 mph (that's f(t)), then you would only get halfway there. So you'd have to do it twice. If you have a constant inside an integral, you can pull the constant outside of the integral to get the integral from a to b of Cf(t)dt is the same as C * the integral from a to b of f(t)dt. So you can either go fast or you can go slow, but do it two times.

The Additive Property
Additive Property

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