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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

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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.

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 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.

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.

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.

Next, we have the 'Keep Going Property', also known as the **Additive Property**. In this case, we have the integral from *a* to *b* of *f(t)dt* + the integral from *b* to *c* of *f(t)dt* = the integral from *a* to *c* of *f(t)dt*. All I've done here is take my velocity from *a* to *b* and find my area. I've then added to it the area from *b* to *c*. That's the same thing as finding the area between *a* and *c*. That would be like if I were on a road trip. After an hour, I look at how far I've gone, and then at the second hour I look how far I've gone since that first hour. It'd be the same thing as if I had looked at the whole two hours I had been driving. So the integral from *a* to *b* plus the integral from *b* to *c* is the same thing as the integral from *a* to *c*.

Lastly, we've got the **Sums Property**, or 'Fast Lane Property'. Let's say that you're finding the integral from *a* to *b* of *f(t)* + *g(t)*. Let's say *f(t)* is how fast the traffic is going on the freeway, and *g(t)* is how fast you're going on top of that. So your total speed is the speed of everyone else in traffic plus the difference between your speed and everyone else's (how fast you're pulling away from everybody). So the integral from *a* to *b* of (*f(t)* + *g(t)*)*dt* = the integral from *a* to *b* of *f(t)dt* + the integral from *a* to *b* of *g(t)dt*.

Let's review the **properties of integrals** that we've learned:

- The
**Zero Integral Property**is the integral from*a*to*a*of*f(x)dx*=0 - The
**Backward Property**is the integral from*a*to*b*of*f(x)dx*= - the integral from*b*to*a*of*f(x)dx* - The
**Constant Property**is the integral from*a*to*b*of*Cf(x)dx*=*C** the integral from*a*to*b*of*f(x)dx* - The
**Additive Property**is the integral from*a*to*b*of*f(x)dx*+ the integral from*b*to*c*of*f(x)dx*= the integral from*a*to*c*of*f(x)dx* - The
**Sums Property**is the integral from*a*to*b*of (*f(x)*+*g(x)*)*dx*= the integral from*a*to*b*of*f(x)dx*+ the integral from*a*to*b*of*g(x)dx*

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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

- Go to Continuity

- Go to Series

- Go to Limits

- Summation Notation and Mathematical Series 6:01
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- Definite Integrals: Definition 6:49
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- The Fundamental Theorem of Calculus 7:52
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- Go to Area Under the Curve and Integrals

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