Evaluating Improper Integrals

When we learned about definite integrals, we saw that we can evaluate the antiderivative over the limits of integration to get a number, the area under the curve over that interval. But what if that interval is infinitely large? Rather surprisingly, an infinitely large interval can actually yield a finite area! Let's see how this works by checking out a variety of different improper integrals and attempting to evaluate them.
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[@chris2656]

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[@navjotsingh2251]

These videos are effective because: A) they are simple B) they use nice animations for demonstration C) he shows the necessary formulas This is something a lot of math lecturers fail to do when teaching.

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[@kiwisorbet]

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[@Divine_Discoveries]

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@@Divine_Discoveries haven't graduated yet, but I'm in my second year into my electrical engineering bachelors

[@egelsia7009]

Hey Proffessor, I have an exam next week and you made me understood nearly all of the subjects. Thank you for everything, have a nice year ♥

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[@anysianas5099]

You make me love maths more and more your explanations are so clear easy to understand prof Dave thank you 🙏🏾

[@daisyg509]

Thank youuuu. I have seen this in calc 2, audited calc 2 and still didn't understand UNTIL I watched this video. This was one of the few calc 2 concepts I did not understand well.

[@kimberlyhoang5191]

i like how you went in depth about certain parts. It was very easy to understand.

[@bernardnyarkoh1798]

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[@IronMan-di3zi]

Sir your explanation is great. I especially liked the way you taught here with geometrical interpretation which I was looking for ❤️❤️ Thank you sir for this amazing content

[@heeberman]

Loved your explanation for the arctan limit. This was a confusion I had from a lecture last night and I just stumbled upon this in your video serendipitously.

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If you pay him more regularly he has more options to buy time for making videos more regularly. Or we can make it possible for him and why not for everyone to get free access to all the necessary resources.

[@JesusMartinez-zu3xl]

wow! You explained this concept better than my cal instructor in 13 minutes! Thank you!

[@Pochix2024]

You are truly a savior Prof Dave♥️

[@gilesrobinson4066]

This explanation was a lot better than the one at my uni, thanks

[@salmaelabsi]

thanks professor dave you're the best!

[@bathtubanarchy]

6:10 I don't recall seeing this common integration mentioned before?

[@payalsagar1808]

crystal clear explanation🙌

[@graced4844]

subscribing because even though my calculus prof teaches fine, we go way too fast and i always need help. thanks dave!

[@schifoso]

Excellent examples.

[@goducksgo613]

Yes, the one where the area was pi blew up my mind

[@ilovecalculus2807]

Yay! Thank you very much.. May God bless you and your family.

[@katherynew]

Excellent simplification of this rather nebulous concept complicated by wordy textbooks!

[@asema.1484]

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[@veliamomo6561]

Great Teacher, thank you

[@OwlFatherTarnished]

Bro an absolute life saver

[@yagzyalcntas553]

how about partial integration for improper integrals? the uv part is calculated using the limits? or just left as it is?

[@codexhd8689]

Nice explanation. Watching from zambia

[@oshawastaken]

DAVE U ARE A GODSEND

[@esramazlan7445]

Thanks sir

[@derplerp8412]

I have an exam tmr and my teacher is not the greatest but because of you I might do well now

[@suyogadhikari2349]

This video is amazinggggg!!

[@vaiterius]

Who made your intro sequence? it is legendary

[@ian.ambrose]

He made it himself. He is an animator too.

[@brunomartel4639]

in 5:14 you forgot to put the limit of t that tends to inf in the rhs

[@michelsupply5328]

SIr Love from India,,,,,,,,,,,,,, Sir i was doubt on this topic but still i understood perfectly...............

[@nirajkc224]

So cool professor

[@issacoh8675]

Quick question: At 9:34, shouldn't the bounds be changed since you are using u-substitution. I know that they will change back after that step, but for that specific step, shouldn't the bounds be different or are they supposed to stay the same?

[@michaellee7778]

The bounds should be different. The upper bound will become 3 (since x - 2 = u , 5 - 2 = 3) and the lower bound becomes t - 2.

[@calebknight8213]

I believe the reason he didn’t change the limits of integration is because he substituted x-2 back in for u after taking the integral. Normally with u-sub on definite integrals you don’t plug your original substitution back in, so you need to change the limits with respect to u. Because he changed his function back to x values, there was no need to change the limits of integration.

[@LaGGSBD]

8:13 couldn't you get 360 as the result if you used degrees instead of radians?

[@ProfessorDaveExplains]

we always use radians in such a context because pi is a value that makes sense in terms of an area whereas degrees do not. also, pi is 180 degrees anyway.

[@zeinabmohamed1668]

استفدت جدا الحمد الله

[@bikram_kumar]

We can use the L'Hospital's Rule for evaluating the limit, instead of checking the graph for the limit values ;)

[@SurajKumar-ey2et]

5:15 i didn't understand what he meant by saying " One over x squared gets small enough fast enough that the area under the curve is finite, while one over x also gets small enough, as it too, goes to zero as x approaches infinity, it just doesn't do it fast enough "

[@utilizator1701]

9:36 you have forgotten to change the limits of the integral (it should be from t-2 to 3).

[@dangyo962]

true but it doesn't matter because he switched u back into x - 2

[@jursamaj]

It should not be surprising that the integral of 1/(1+x^2) is convergent. Over the span from 0 to 1, it clearly has a finite area. Meanwhile over the span 1 to ∞, it is always less than 1/x^2, which is convergent.

[@acutespike9260]

btw how can we know that we should use improper integral without drawing the graph? :D

[@kris-yg9om]

Revolutionhk4 when you need to calculate the area between an asymptotic value and a real and definable x value, same applies to the y axis if you function Has horizontal asymptote

[@flamingturkey7727]

Use this method whenever there’s an improper integral. Essentially, it’s improper whenever the bounds include infinity or negative infinity or when there’s a vertical asymptote (you should know from many other math classes that if there’s an x value that would make the denominator 0 then there’s a vertical asymptote there.)

[@s-semane]

i have many questions anyway!

[@Schlohmotion]

11:33 dx/x^4 .... does this stand for 1/x^4 dx ?

[@ProfessorDaveExplains]

yep same thing!

[@Schlohmotion]

@@ProfessorDaveExplains Professors seem to answer faster than teachers, neat! Thank you.

[@edison9581]

Hi professor, how come the solution of tan^-1(0) is not 0 + n(pi), but is just 0?

[@luminousvalentine8011]

I think that's because we only take the minimum values and if we write it as n(pi) it might complicate the results

[@bobthebb-illder827]

8:28

[@jumanahabiballa6258]

Thankss Dave 😏

[@hellboy4167]

You are still saving your students degree😭💯

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I love this guy 😝

[@brxyann]

Legend

[@matemaatika-math]

I hope that this time you wouldn't argue that there's a part that needs to be corrected at 11:39: You must put parenthesis around 3 * t ^ 3 as otherwise, t ^ 3 would be part of the nominator or whatever you English-speaking crowd call the fraction part or the part that's on top of the fraction line. According to your notation we can read -1 / 3 * t ^ 3 === -t ^ 3 / 3.

[@Vagdebrume]

You're still awesome :)

[@Vagdebrume]

And video is a really good way to teach maths :)

[@elamvaluthis7268]

Very nice explanation 😂🤣🤣😄🙏👍.

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nc sir

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sir you are from which country?

[@ProfessorDaveExplains]

USA

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i love you

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just mazing:)😘

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Here for a math calc 1 uni exam

[@rojaponduru4106]

Please say easily

[@ProfessorDaveExplains]

this is as easy as it gets, bud!

[@matemaatika-math]

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[@apurbasharma]

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[@dailyrosesofholyquran780]

👍

[@SurajKumar-ey2et]

5:15 i didn't understand what he meant by saying " One over x squared gets small enough fast enough that the area under the curve is finite, while one over x also gets small enough, as it too, goes to zero as x approaches infinity, it just doesn't do it fast enough

[@ian.ambrose]

Thank you Jesus Dave Professor.

[@GavenYurisich-nu5zn]

A bandai

[@dennisemacapagal4331]

From chemistry jesus to mathematics/calculus jesus

[@matemaatika-math]

Here at 6:18, you're using ambiguous notation again as also Krista King does however I remember her mentioning that one must be sure of the context before reading. So if you use -1 for arcustangens then your saying can be interpreted as cotangens as well. I suggest to write directly arctan and not using the inverse notation. I suggest arctan and even not atan as I've seen you using atan earlier in the meaning of a * tan as you told me that multiplication was implied on adjacent terms. How in our holy green world can we know that you mean tangent and not three terms t, a and n which are multiplied together? Because nobody past like fourth degree of something thinks that way?

[@davethesid8960]

But pi isn't finite! 🙄

[@ProfessorDaveExplains]

Um, yes it is.

[@davethesid8960]

@@ProfessorDaveExplains Wow, thanks for replying, I was just saying that although it IS a finite value, it cannot be written out at its full capacity... Eh, as kind of a joking around

[@ProfessorDaveExplains]

Ah, yes, it's irrational.

[@davethesid8960]

@@ProfessorDaveExplains And also thanks for such a clear explanation about integration. Your way of teaching is just sooo different (in a positive sense) from plain old scholar education, I love it! ❤

[@davethesid8960]

@@ProfessorDaveExplains Actually I'm so keen on maths that I decided I would like to become a maths professor one day too, hope I'll achieve this goal.

[@matemaatika-math]

I've found yet another ambiguous notation, this time a verbal one at 7:38. It's hilarious that we get a whole pie altogether however what we get here, isn't a pie but it's a something that you represent with a Greek letter that will be pronounced as "pee", so half-pee and half-pee are a whole pee and if we want to illustrate that area we could color is after pee as it's full of pee. But whose pee is blue? Because it's a blueberry pie.

[@IbrarKhan-fb9ze]

Thanks sir