The Trapezoid Rule for Approximating Integrals

Approximating Integrals Using the Trapezoid Rule
In this video, we'll explore how to use the trapezoid rule to approximate the value of a definite integral. We introduce the formula and walk through a basic example to help you understand how this numerical integration technique works. A great resource for anyone learning calculus and looking to grasp one of the methods for approximating integrals.
What You Will Learn
The formula and concept of the trapezoid rule for numerical integration.
How to apply the trapezoid rule to approximate a definite integral.
Step-by-step calculations using a simple example.
How to interpret the results
The trapezoid rule is a practical technique for approximating the area under a curve when finding the exact integral is challenging or impossible. In this tutorial, we break down how to use the rule by dividing the interval into subintervals and applying the trapezoidal formula. By the end of this video, you'll be comfortable using the trapezoid rule to approximate definite integrals and understand how it compares to exact integration.
If this content helps enhance your understanding, please like, comment, and subscribe! Share it with classmates, teachers, or anyone interested in learning calculus techniques.
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[@Dean444ful]

Why can't professors be like this? I swear, in the first 4 minutes of this video I managed to understand how the trapezoid rule works. My professor has spent about a total of 30 minutes trying to explain it to the class and no one gets it. What the hell? It's actually pretty simple. Idk if professors and high school teachers take some sort of test to see if they can actually teach well but they should. It's insane how many bad professors there are.

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[@patrickjmt]

the playlists are a pain to keep organized; there is a lot of stuff missing from the playlists that i have covered. always do a search! :)

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Uploaded 12 years ago and it's still useful today.. Thank you

[@sylvainbarbot4104]

Nice job on the video. Also, it is worth commenting that the exact solution of the integral is 136/3=45.3333. So the approximation with n=4 trapezoids is pretty close. Of course, the higher number of trapezoids will give a more and more precise approximations. As a limit, using n=infinity, the approximation will be exact. This is in fact one possible definition of the integral operator.

[@vyellaa]

I am able to understand! My professor went on a huge ramble about this and I understood nothing out of it. Thank you!

[@inuyome1008]

I've been confused with what the multiplying factor in front of the sections should be, but I think you cleared it up! Thanks so much!

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What I really like about your videos or really any type of learning not in a classroom, is that the teacher doesn't have to constantly stop and say some variant of shut up. But even better about your videos are that you're serious and confident, but have a joking manner. i don't know if that makes sense, but whatever.

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Cut my function into pieces, this is trapezoid rule

[@smuganimeface1116]

I tried

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smh

[@vladp6323]

Integration, no reason, for me to know this due to prevalence of software in engineering.

[@darkcnotion]

thank you

[@patrickjmt]

@SuperArglebargle btw, this is so easy to derive if you were not so mindless and thought about it for like .1 seconds. i guess you need to be spoon fed all things. i used to have a teacher and he would tell students they were like little babies. now i know what he meant!

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[@Thfro920]

@patrickJMT I figured it out but what I meant was essentially the proof of it that leads to the simple formula. It's basically taking the area of each individual trapezoid. The proof then distributes it to bring it to the normal trap rule. I asked my question poorly.

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you get the exact answer

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@JDMimportsFTW it is part of the reason i do not teach any more (but only part)

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@chaoskie the formula is actually DeltaX = (b - a) / n n being the number of strips or rectangles u use

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You're learning this in Calculus AB? I thought that class is the equivalent to college Calc I where you only cover Derivatives and some Integrals.

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[@patrickjmt]

thanks!

[@xgraysunx]

Assuming H is delta x and Y is the function, this is the same trapazoidal rule, the only difference is you add the first and last term first, they all the terms in between and multiply them by two. Just look at the very first thing on the video, you should be able to see how they got it.

[@patrickjmt]

could not help being a bit of a smart ass. that is just how i roll.... i know, i was just too lazy to figure it out. thelichking1 suggests it is 45.33, which i am going to agree with. thanks LK1!

[@jhetao]

It depends. Its a great full, college course and is obviously very difficult. However, BC Calc should have prepared you for it. The price for the class can vary. For example, in Wisconsin, a law forces my school district to let me take it for free if I'm a junior or senior. So thats a >$1000 course for free.

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[@patrickjmt]

@Thfro920 you dont do it with unequal intervals.

[@patrickjmt]

@1337Bigman there is something you are missing.

[@94cd5]

I missed the day we covered this in class, thanks for the video. It was easy to understand.

[@ngandjuicynthia9168]

thnx for the video,i was soo confused wit the trapezoid rule b4 i watched dis.

[@roderik1990]

@patrickJMT that is correct, the exact answer is 45+1/3

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[@minavasquez7691]

Totally missed the lecture for this concept....made it too easy thanks

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Your video is awesome! Obviously you did a simple example considering you only used n=4. We are learning to use the sigma notation because we are dealing with n=50 but we are also using the calc and as a JMT fan you hardly EVER use calculators which may explain why you didnt do one like that. However I was hoping to find a more difficult one. Class ex was Intergral from 0 to pie/3 of cos^2(x)

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Preparing for the final... this helps. Thanks!!

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[@prcushinplaya08]

hey patrick, appreciate the video, but is there a reason it isn't in your integral playlist? I tend to go through your playlists as I review, and not seeing this topic, I thought you simply didn't cover it. Whatever the reason, great vid, thanks.

[@Admin-wf4jx]

Perfect example. Thank you,

[@patrickjmt]

@MrSnailshit that was a long time ago. i dont use it any more

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[@ICOD73]

In order to find the true error whether using single application or multiple application of the trapezoidal rule we just use the the formula: True Error= ABS[(True-Aprrox)/True]*100% .......... Am I correct?

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[@patrickjmt]

@JDMimportsFTW i make a lot of money doing it full time, but thanks for the recommendation : )

[@MrMoeqt]

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[@tenken1337]

omg 46/2 = 23. It's EVERYWHERE!!! =P but seriously, thanks for the video. We were just about to learn this tomorrow and I really didn't understand what my book was saying. Keep up the good work! =)