Double integrals using polar coordinates

  • Thread starter popo902
  • Start date
  • #1
60
0

Homework Statement



Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles
x^2 + y^2 = 4
and
x^2 - 2x + y^2 = 0

Homework Equations





The Attempt at a Solution



for my integral i got
0<= theta <=pi/2 for the theta bounds since it lies in the 1st quad
and i got
2costheta<= r <= 2 for the bounds of r because 2costheta is the polar version of
x^2 - 2x + y^2 = 0 and 2 is 'r' in the equation x^2 + y^2 = 4

and the equation that i integrated is (2- 2costheta)r
you add the extra r when turning the equation polar and the equation r=2 is over
r = costheta when i graphed it
and welll...
i got pi - 3 but it's wrong...

am i even setting up the equation right or am i just having problems with the math?
 

Answers and Replies

  • #2
35,392
7,269
This is a double integral, so what you want is this:
[tex]\int_{\theta = 0}^{\pi/2}~\int_{r = ?}^{?} r~dr~d\theta[/tex]

IOW, the limits of integration describe the region, and all you need for the integrand is 1dA. You seem to have a handle on how to describe the region, so use what you have found for the limits for r.

I get [itex]\pi/2 - 1/2[/itex] for the area.
 
  • #3
60
0
r u serious??
why is it do i integrate 1dA?
i saw that in the book for finding the area of one of the clovers of a cosine function
but it didn't explain why i just integrate 1dA
 
  • #4
35,392
7,269
Yes, I'm serious. For a region R, this double integral gives the area:
[tex]\int \int_R 1 dA[/tex]

In rectangular coordinates, the iterated integral will look something like this:
[tex]\int_{y = a}^b \int_{x = f(y)}^{g(y)} 1 dx~ dy[/tex]

and dA = dx dy or dA = dy dx. If you integrate in the reverse order, the limits of integration will be different.

In polar coordinates, dA = r dr d[itex]\theta[/itex]. That's where the r came from.
 
  • #5
60
0
oh right. I'm looking for the area.
double integral of 1da gives the area and triple integral of 1 gives the volume of a 3d object.
thank you
 
  • #6
35,392
7,269
oh right. I'm looking for the area.
double integral of 1da gives the area and triple integral of 1 gives the volume of a 3d object.
thank you

double integral of 1dA gives the area and triple integral of 1dV gives the volume of a 3d object.
 

Related Threads on Double integrals using polar coordinates

Replies
6
Views
1K
Replies
3
Views
3K
Replies
7
Views
1K
Replies
2
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
2K
Replies
3
Views
1K
Replies
3
Views
13K
Replies
3
Views
831
Top