Integration: Area Enclosed By Graph Of $X^4 + Y^4 = 1

I"m attempting lớn find the area enclosed by the graph $x^4 + y^4 = 1$ as shown below.My approach was to lớn rearrange the equation so it is in terms of $y = f(x)$ và integrate one of the đứng đầu two quadrants with respect to $x$ & then multiply by $4$ lớn get the area for the whole shape. I"ve never tried to lớn integrate this kind of graph before và I"m not sure If I"ve done it correctly. Any input đầu vào or assistance would be much appreciated. Thanks.

Bạn đang xem: Integration: area enclosed by graph of $x^4 + y^4 = 1




$egingroup$ you are wrong. I dont think $int (1-x^4)^1/4 = frac45(1-x^4)^5/4 +C$ $endgroup$
Essentially the solution you have is good, but I would lượt thích to have this one anyways.

The required area is by symmetry

$$ A = 4 int_0^1 (1-x^4)^frac14 hspace4pt gamize.vnrmdx$$

Substitute $u = x^4 hspace4pt Rightarrow x^3 = u^frac34 hspace4pt, gamize.vnrmdu = 4 x^3 gamize.vnrmdx $

$$ eginalign*A &= 4 int_0^1 frac(1-u)^frac14 hspace4ptgamize.vnrmdu4 u^frac34\ &= int_0^1 u^frac-34 (1-u)^frac14 hspace4ptgamize.vnrmdu\ &= fracGammaleft(frac14 ight)Gammaleft(frac54 ight)Gammaleft(frac32 ight)\&= frac2 imes 3.287sqrtpi \&approx 3.708endalign*$$

tóm tắt
answered May 5, 2012 at 0:31

Kirthi RamanKirthi Raman
7,42422 gold badges3434 silver badges5959 bronze badges
add a bình luận |
In general, the curve


is called a superellipse, and it"s area is given by

$$4cdot Gamma(alpha+1)^2 over Gamma(2alpha +1)$$

Here $Gamma$ is Euler"s Gamma function, defined for $Re(alpha)>0$ as $$Gamma(alpha)= int_0^inftye^-mumu^alphafracd mumu$$A closely related function is Euler"s Beta function, given by

$$B(a,b)=int_0^1 t^a-1(1-t)^b-1dt=fracGamma(a)Gamma(b)Gamma(a+b)$$

for $Re(a),Re(b) > 0$.For the details, you can see this question.

tóm tắt
edited Apr 13, 2017 at 12:20

answered Apr 28, 2012 at 23:59

118k1616 gold badges207207 silver badges368368 bronze badges
add a comment |

Your Answer

Thanks for contributing an answer khổng lồ gamize.vnematics Stack Exchange!

Please be sure khổng lồ answer the question. Provide details and share your research!

But avoid

Asking for help, clarification, or responding to lớn other answers.Making statements based on opinion; back them up with references or personal experience.

Xem thêm: Từ Văn Bản Bàn Luận Về Phép Học Em Có Suy Nghĩ Gì Về Mục Đích Và Phương Pháp Học Của Bản Thân

Use gamize.vnJax to lớn format equations. gamize.vnJax reference.

To learn more, see our tips on writing great answers.

Xem thêm: Top 9 Bài Tóm Tắt Truyện Tấm Cám Theo Nhân Vật Cám Theo Nhân Vật Cám

Draft saved
Draft discarded

Sign up or log in

Sign up using Google
Sign up using Facebook
Sign up using email and Password

Post as a guest

email Required, but never shown

Post as a guest


Required, but never shown

Post Your Answer Discard

By clicking “Post Your Answer”, you agree lớn our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.
Featured on Meta
The area of the superellipse
Area between 2 curves
Finding the area on a graph bounded by four curves.
What is the proper way to set up this integral to lớn find the area bounded by the curves?
Find area enclosed by three functions
Integrand of a double integral
Volume Enclosed Between a Surface and a Plane
Differentiation + integration: how khổng lồ solve for acceleration & displacement given a specific velocity time graph?
Integration: find as an exact value the enclosed area between $y=frac3x5π$ and the curve $y=sin x$ for $0≤x≤π$ shown shaded in the diagram.
Quick Volume of Revolution Question
Quick Circle Cross Sections Question
Hot Network Questions more hot questions

Question feed
Subscribe to lớn RSS
Question feed to lớn subscribe to lớn this RSS feed, copy and paste this URL into your RSS reader.

Stack Exchange Network
Site thiết kế / hình ảnh © 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Rev2022.11.22.43050

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.