Double integral with polar coordinates

  • #1

Homework Statement


It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ.

f(x,y)=1 (plane parallel to Oxy plane)

They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it.


Homework Equations



x=rcosθ
y=rsenθ
r=√x2+y2

The Attempt at a Solution



I've done the variable substitution as:

0≤rcosθ≤1, 0≤rsenθ≤1-cosθ and ∫∫Setrdrdθ

After analysing it for a bit I figured that 0≤r≤1 and that 0≤θ≤[itex]\frac{\pi}{2}[/itex].
However, the solution to the integral is 0.5. For the limits I've established, it gives me [itex]\frac{\pi}{4}[/itex].

I can easily calculate that integral in x,y coordinates but I'm having trouble defining the endpoints of r and θ when changing a set from x,y coordinates to r and θ coordinates.

Can you help me with this?
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770

Homework Statement


It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ.

f(x,y)=1 (plane parallel to Oxy plane)

They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it.


Homework Equations



x=rcosθ
y=rsenθ
r=√x2+y2

The Attempt at a Solution



I've done the variable substitution as:

0≤rcosθ≤1, 0≤rsenθ≤1-cosθ and ∫∫Setrdrdθ

After analysing it for a bit I figured that 0≤r≤1 and that 0≤θ≤[itex]\frac{\pi}{2}[/itex].
However, the solution to the integral is 0.5. For the limits I've established, it gives me [itex]\frac{\pi}{4}[/itex].

I can easily calculate that integral in x,y coordinates but I'm having trouble defining the endpoints of r and θ when changing a set from x,y coordinates to r and θ coordinates.

Can you help me with this?

##r## doesn't go from 0 to 1. In your picture, pick some ##\theta## and draw the ##r## for that ##\theta##. ##r## goes from 0 to the ##r## value on the line. So write the equation of the line in polar coordinates and solve it for ##r##. That is your upper limit on ##r##.
 
  • #3
DryRun
Gold Member
838
4
What's senθ? :smile:

Using your given limits for x and y, you should draw the graph, so you can understand and derive the limits for polar coordinates.
 
Last edited:

Related Threads on Double integral with polar coordinates

Replies
13
Views
469
Replies
16
Views
6K
Replies
5
Views
354
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
12
Views
976
  • Last Post
Replies
7
Views
11K
Top