• Support PF! Buy your school textbooks, materials and every day products Here!

Double Integration (Stuck at square root step) (Solution Included).

  • Thread starter s3a
  • Start date
  • #1
s3a
800
8

Homework Statement


The problem and solution are included.


Homework Equations


Double integration.


The Attempt at a Solution


Firstly, I'd like to mention that the additional ρ under the square root is there accidentally and that it should be outside of the square root such that it forms the ending ρ dρ dθ.

I'm stuck at the sqrt(4a^2 - ρ^2) ρ dρ step. I have a feeling that I need to use trigonometry with ρ = 2acosθ or ρ = 2asinθ but firstly, which of the two would I choose and secondly given that I then have to integrate with respect to θ, must I replace ρ with a variable other than θ? Why or why not?

Any help would be greatly appreciated!
Thanks in advance!
 

Attachments

Answers and Replies

  • #2
60
0
I did it with [itex]\rho=2a \sin t[/itex].
 
  • #3
s3a
800
8
Thanks for saying but I also just realized that I don't know why z = √(4a^2 - ρ^2). Could you explain that to me please?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
956
The "sphere of radius 2a", which forms part of the boundary, has equation [itex]x^2+ y^2+ z^2= 4a^2[/itex]. In polar or cylindrical coordinates (not spherical coordinates), [itex]\rho=\sqrt{x^2+ y^2}[/itex] so that equation becomes [itex]\rho^2+ z^2= 4a^2[/itex] so that [itex]z^2= 4a^2- \rho^2[/itex] and [itex]z= \pm\sqrt{4a^2- \rho^2}[/itex].
 
  • #5
s3a
800
8
How did you go from ρ = √(x^2 + y^2) to ρ^2 + z^2 = 4a^2?
 
  • #6
60
0
Change [itex]\rho=\sqrt{x^2+y^2}[/itex] in [itex]x^2+y^2+z^2=4a^2[/itex]
 

Related Threads on Double Integration (Stuck at square root step) (Solution Included).

Replies
5
Views
1K
Replies
1
Views
1K
  • Last Post
Replies
3
Views
911
Replies
1
Views
4K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
13
Views
15K
  • Last Post
Replies
8
Views
56K
  • Last Post
Replies
21
Views
2K
Replies
3
Views
1K
Top