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Consider the integral $\int_{0}^{\pi} \sin x d x$

(a) Use a calculator program to find the Trapezoidal Rule approximations for n = 10, 100, and 1000.(b) Record the errors with as many decimal places of accuracy as you can.(c) What pattern do you see?(d) Writing to Learn Explain how the error bound for $E_{T}$ accounts for the pattern.

a) $T_{10} \approx 1.983523538, T_{100} \approx 1.99983504$ and $T_{1000} \approx 1.999998355$b) 1000$\quad 1.645 \times 10^{-6}$c) $\left|E_{T_{10 n}}\right| \approx 10^{-2}\left|E_{T_{n}}\right|$d) $\leq 10^{-2} \cdot\left|E_{T_{n}}\right|$

Calculus 1 / AB

Chapter 5

The Definite Integral

Section 5

Trapezoidal Rule

Integrals

Integration

Harvey Mudd College

University of Michigan - Ann Arbor

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Boston College

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we want to use some kind of ocular program to help us evaluate the integral zero. The pie of sign of excusing the trump is over rule when it is able to 10 100 1000. And then we want to record the errors with as many decimal places as Acrisius can. And then, lastly, we want to see if we can find a pattern and then finally explain this pattern using the ever bone formula or the trump is idle method. Now, if you just have a calculator that can plug everything in, you don't need to do everything that I'm about to show you how to do an XL. But in the case, you don't others to inging about this. If you don't have just a calculator that will do, trap is a little method right away. And one is by looking at this as a re mont. Some using this formula here and also recall that H is going to be B minus a over. Yeah, well, let's go ahead and figure out what age should be in each case. So in the case of t 10 h is going to be over integrated from zero to pie, so it should be pi minus 0/10. We get high, Tim. So if we want to find t 10 this here is going to be the sum of So the pie over for pie tin over 20. Sum from I is equal to zero two in minus one, which is gonna be nine of. And unless for what? X I should be on. We can do that off on this side right here. Remember X. I should be eating too. A breakfast is in top right corner. So remember that X I should equal to a plus by times h But in this case, a is zero. So that goes away. And then we could just plug in H, which we found to be pi over 10. So this should be I pi over 10. Now, let's plug this in for each of our values here. There's gonna be Oh, aye, aye. Over too. And then plus, yeah, uh, I plus one times pi over Tim since this year is X I in this Syria's Excite Plus one and effort bags this sign of X. Let me write that in green. So this here is f of X. Let's go ahead and just replace AF here with Cy. That's too big. So sign. Sorry. Now, if you were to plug this into a calculator that has remained some banksability, then that will give you the answer. Let's go ahead and set up the other ones as well. And if you don't, I can show you how you can do this using excel so we'll just go through these little bit quicker. So in this case, it should be high. Over 100 which would give us t 100 is equal to. So it's high over 100 divided by two soap I over 200 announced. Could be son from Isaac Go to zero up till 99 because I would be one minus in and then we would have signed, uh, I times pi over 100 and then plus I sign of I plus one high over 100. Because remember, we're using this this formula right here and we're just plugging in for H. And in each case, a is still zero. And so then, if you were to plug this one in to some kind of remind some calculated that would also give you tea 100. And then lastly, we're Team 1000. Just give us pi minus zero over 1000. See the tubes, Opie. Over 1000. That gives us Team 1000 is equal to high over 2000. Now, this is son from Isaac zero 2 99 for 999. A sign of Ai pi over 1000 plus sign of I plus one high over 1000. So looking all those three and should give you the area or approximation of our area. So let's go ahead and see what I got when I plugged all this into itself. So just kind of explain it for N is equal to 10. So that would be This is NBC 10 and is able to 100 is into 1000. This top part here was just helping us solve For what? Our value of h is, um, this column of ends here is just for placeholders. Um, just in case, if I would have plugged something in weird and it went past the value I was looking for, I at least know I needed to stop at it and see 10. The next column gives just our input values. So that would be like a plus h plus age plus age. Plus age is going all the way down like that. Ah, than F of X is just taking whatever value we had here, the sign of it. So that's how we would get that calm of FX is and then to get area column What I did waas I added these two cells together, divided it by two. And then I just repeated that so like, this would be I is equal to zeros area. Then the next one here with the eyes that one all the way down to here I is equal to 10 and I just repeated that and then to get this total area, I just summed up this entire call him here. So that's what I got. Uh, and then for any 0 to 100 I got this area the 1.9998 and then for any secret 2000 I got 1.999983 by five. And I just have these values down here for each just so if you want Oh, double check with what I did if you set this up in Excel that at least er it values should match Oppa's Well, now they say we want to then in part, be figure out the error. And so we should probably say how we got this equation here, that the error is to minus Tien. Let's go back over to here. We'll figure out what this value of sign actually is. Remember, we would just go ahead and integrate it. And this is going to be co sign XVIII Wade from zero to pie buggin end. Hi would give us I'm sorry. That should be negative Coast. So plugging in pie would give us negative ones. We have negative one times one and then minus when we plug in one, which would be just negative one. So the negatives in each case cancel out and we end up with an area of two. So the actual area and B do, and then we just subtract it from what we got from each of these right here. So that's how I got these three values. And they asked for us to find a pattern in the error and in our case, noticed that between each of these. There's a multiple of 10 to the negative second cause. Remember that E is just saying times 10 to the power of whatever follows. So the pattern is 10 to the minus two multiple to get next, ever. And this is just when in increases by a factor of 10. Then they asked us to use the actual air about formula to describe this. All right, now what I'm going to do is I'm just going to collect these two things together because for all of ours, neither of those should change what I highlighted in blue. So I'm just gonna be be minus a over 1 80 I'm sorry, that should be over 12 1 80 is for Simpsons. So over 12. And I'll fix it right here as well. Over 12. And then we have m the double prime Now, h remember, it's supposed to be B minus a over in squared. Okay, so we can go ahead and move that b minus Say to be my to say, cubed over 12 in double prime, and then we just have one over end squared. And in each case, if we increased by a factor of 10 than end would increased by 100 says We're plugging in 10 and it's just getting square. To become a hundreds of this right here is why we would expect that pattern we saw in parts.

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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