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The graph of a function $ f $ is given. Estimate $ \displaystyle \int^{10}_0 f(x)\, dx $ using five subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints.

(a) 6(b) 4(c) 2

00:03

Frank L.

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Campbell University

University of Nottingham

Idaho State University

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All right. We've got a question here where we have the function of f. And the graph, given you want to estimate the integral from 0 to 10 For the function of X. using 5 similar rules With part a being the right endpoints Or B being the left and see being in the points. So here we know that we have the interval from zero to tense. We will be able to calculate or change in X As our larger value uh minus our Smallment values. What about 2? And here are being a is a representation of the interval by the way? Or B would be the larger end of the interval, which is 10 and then 0 is the smaller end of the interval divided by 2. The change in X is equal to what Students. Sorry, I did that wrong. The bottom part here is the number of intervals just end And the number of intervals we know would be 5. So that you will change in access to. All right. Now When we calculate for our um our excuse me When we calculate for that integral Using the right endpoints, what we're essentially doing is we're Writing out the Riemann sum Which is the change in X just to Multiplied by the function of Aeg all of your sub intervals from 0-10. And we know we have 57 roll so you could have a function of X. Someone Was all the way to the function of X sub plot. Okay. And so These answers in here are going to be When you have to 46, 8, 10. Okay. Because we know that it's changing The change of x occurs every two or are changing taxes too. And we have sides 5/7 levels. So basically what we do here is we say what is a function of x upon which was to Exa 1 is to what's the function of excellent access to? So you would see that X is equal to to the function of X here is equal to negative one. Okay. In the function of X two, X. Up to before at four you have Where you have also in negative one. I'm sorry. No, this is for him. 1, 2, 3, 4, 4. So it's actually zero. Okay. And then at 6 we have here negative too. And then at 8 we have positive too. and then at 10 we have a right The right. Is that 3 years at 4? That's right. Okay. All right. Now let's go ahead and solve for this. So we have a negative one plus negative two which is negative three negative three plus two is one plus 433 times to six. All right. Now, when we saw for parts B and C, once again, we're doing the same thing. We're using remained some but we're now we're doing the left end points and the left hand points. All it does is it keeps the entire format. But instead you start off with an X sub zero In your X. of 0 would be too Right out your left, some be equal to two times at sub zero. So the aftereffects of zero one Plus all the way to your Activex 5, I'm sorry, 4 As we can only have a total five intervals. All right. So you would say, well what is the value at X zero? We know except 00 So what is the function of X. X is equal to zero and zero? We have here three. So we would write out To mortify those euro And the x of one is to which we know is negative negative to negative one. and then accept to is I'll be here in that 0 And then plus we have a negative 2 again and then finally at X sub. Or we have to And when we add all these values together we have a negative one multiplied by two. These cancel out. So you're left with -2. This was supposed to be 3. I'm sorry. Right because X is equal to zero. We have a three here. So it's really 3 -1 of just 22 times two is four. All right. And the way that you find part C for mid points Is you basically take the interval the The interval between your sub interests. Okay. So I guess to explain that further You would take the value in between X0 and X one X of one. And you just keep doing that up until you get your five intervals. So let me write that out to make it clear as you have em equal to two, multiplied by the average or the Midpoint between except 0 and your ex 1 you would say well what is the value between X zero next to? That's the one which we know X. Of one is two and X zero. This is your except one, this is your except zero. So the middle value would then be at one and we can see that one a function of X0. And then you would just keep doing that so on and so forth. So you would choose the value here and we know that there you have a negative 1 and then you go for your next value here, you know right here is the middle between X two and X three and that's also a negative one. You do that again. Here, you would say this value here is 0 And the value here is the word Okay? And then what you're left with is 82 times All of this added together is 1. So it's just too Right. And those are your final answers. Or when you calculate for the integral of your function of X from 0 to 10 Using your 5 sub intervals, which was your Party, you're right endpoints for partly your left endpoints and report see your mate points. All right. Well I hope that clarifies the question there. Thank you so much for watching.

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