Double Integral of Pythagoras over rectangular region

  • #1
Take any given point on the perimeter of a (A x B) rectange and then draw a line from that point to another point on one of the three remaining sides of the rectangle. What is the average length of the line?
Well, the answer to that question involves integrals like this:

[tex]\int_0^A \int_0^B \sqrt{x^2 + y^2} dy dx[/tex]

This thing gets a bit interesting given that it's basically a polar type integral but with rectangular boundaries. Any ideas?

Thanks!

Chris
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
963
That looks like a fairly standard "trig substitution" integral. To integrate with respect to y initially, let [itex] y= x tan(\theta)[/itex] so that [itex]\sqrt{x^2+ y^2}= \sqrt{x^2+ x^2tan^2(\theta)}= x\sqrt{1+ tan^2(\theta)}= x sec(\theta)[/itex] while [itex]dy= x sec^2(\theta)d\theta[/itex]. The first integral becomes [itex]\int_0^{arctan(B/x)} (x sec(\theta))(x sec^2(\theta)[/itex][itex]= x^2\int_0^{arctan(B/x)} sec^3(\theta)d\theta[/itex]
 
  • #3
3,812
92
Take any given point on the perimeter of a (A x B) rectange and then draw a line from that point to another point on one of the three remaining sides of the rectangle. What is the average length of the line?
Well, the answer to that question involves integrals like this:

[tex]\int_0^A \int_0^B \sqrt{x^2 + y^2} dy dx[/tex]

This thing gets a bit interesting given that it's basically a polar type integral but with rectangular boundaries. Any ideas?

Thanks!

Chris

Hi ctchervenkov! Welcome to PF!

Polar works here. You will need to split the region into two.
$$\int_0^{\tan^{-1}\frac{B}{A}} \int_0^{\frac{A}{\cos\theta}} r^2\,dr\,d\theta+\int_{\tan^{-1}\frac{B}{A}}^{\pi/2} \int_{0}^{\frac{B}{\sin\theta}} r^2\,dr\,d\theta$$
Not sure if this can be simplified further.
 
  • #4
to Pranav-Arora: I had tried that, but the things got a bit ugly...

to HallsofIvy: Had not tried that particular substitution I don't think. Thanks. Will see if that simplifies things...
 

Related Threads on Double Integral of Pythagoras over rectangular region

  • Last Post
Replies
9
Views
16K
Replies
5
Views
650
Replies
7
Views
812
Replies
5
Views
637
P
  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
2
Views
1K
Top