Calculating Areas by Integration: Simple Problems
Вставка
- Опубліковано 31 тра 2024
- 0:00 Video Starts
0:26 Problem (a)
2:00 Problem (b)
3:21 Problem (c)
A simple problem on calculating the area by integration. The only integral formula I use in this video is the "power rule," which is the integral of x^r.
The video about the GENERALIZED version of this problem is now uploaded. Check here: • I Wasn’t Going To Gene...
#integration #calculus #areas #integral #AreaUnderACurve #PowerRule
----------------------------------------------------------------------------------------
CORNERSTONES OF MATH features quality math problems to strengthen your math fundamentals and problem-solving ability. Problems are generally on high school level (with some deviations), spanning over topics such as algebra, discrete mathematics, calculus, geometry, statistics, trigonometry, etc. I hope that this channel provides some intellectual pleasure and make you appreciate the beauty of math itself.
Please consider giving a Like to this video and Subscribing to my channel, it really means a lot for the creator like me, and you will be introduced to many more interesting math videos!
The video about the GENERALIZED version of this problem is now uploaded. Check here: ua-cam.com/video/KAYdQu0FJHc/v-deo.html
You can generalise this nicely by using a surface integral instead of a single integral, and by making the substitution u = x^(1/n), y = v^(1/n). Then the Jacobian is n²×u^(n-1)×v^(n-1), and you can integrate with 0 ≤ v ≤ 1 and 0 ≤ u ≤ 1-v.
Either way, you get that the nth area is n×Beta(n, n+1).
Yeah, thanks. I initially didn't thought on covering the general version due to it being somewhat advanced, but seeing that this video is gaining popularity, perhaps I should consider it......
These can be generalized to Int_0^1(dx(1-x^(1/n))^n)=Gamma(1+n)^2/Gamma(1+2n)
A fascinating general expression! However, I don't know if I will cover this topic in near future, because it is bit too advanced compared to the general level of math covered in my channel. Still, thank you for your contribution!
which is also 1 divided by the nth central binomial coefficient :)
@@blub232324 If n is natural number, then yes!