Equation of Graph in Polar Coordinates

In summary, Salman was looking for the area between two circles and wasn't sure how to find the region. He was able to find the area using the double integral.
  • #1
salman213
302
1
1. The question was find the area between the curves using DOUBLE Integrals

Area between:
r = sin theta
r = cos theta


well to draw them i made them into cartesian form by

r^2 = rsin theta
r^2 = rcos theta

so

x^2 + y^2 = y

x^2 + y^2 = x

completing square

1) x^2 + (y - 1/2)^2 = 1/4
2) (x - 1/2)^2 + y^2 = 1/4


these are two circles

their intersection or bounded region that we need to find the area is like a disc...

I know if i use the double integral

Integral of (Integral of 1) dA OVER D where D is the intersection bounded area


i get the area i need...


but I don't know how to define that region

if someone can help me define the region it will be helpful

I know THETA will change from 0 to pi/2

but R will change from 0 to some equation of that area but i don't know how to find that equation!

HELP!
 
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  • #2
Hi Salman

If you look at r = sin theta in polar coordinates, I assume you mean the the polar equation is:

r = (sin(theta), theta)

which is defintely not a circle, similar for the cos
 
  • #3
so

in polar coordinates
r = sin (theta)

is NOT equal to
r^2 = rsin (theta) = x^2 + y^2 = y??
 
  • #4
Hi salman, sorry jumped the mark on that one, think I know what you are doing now...

so you have 2 circles radius 1/2 at centres (0,1/2) and (1/2,0), and need to calculate the area between them

If you draw a picture where do the circles intersect? at the orgin, and at a point along y = x (or equivalently theta = pi/4) due to the symmetry of the situation

think about the bounds for r in each half of the area, both upper & lower will correspond to one of the circles,

I would try and write it as sum of 2 integrals, one from theta 0 to pi/4, and theta pi/4 to pi/2, with appropriate r bounds. So do the r integral first & define lower & up bounds in terms of theta for each segment...
 
  • #5
so in other words i can just do


theta limits 0 to pi/4
r limits r = sin (theta) to the intersection point


add that to


theta limits pi/4 to pi/2
r limits intersection point to r = cos (theta)


would that be the proper limits?
 
  • #6
lookin good - so the first half is

[tex]\int^{\pi/4}_0 d\theta \int^{sin{\theta}}_{0}dr (?)[/tex]

what is your integrand for area? do you know what a jacobian is?
 
  • #7
yea got it ty...it was also on my quiz yesterday so it was great i asked here :)

r*dr*d(theta) :)
 

1. What is the equation of a graph in polar coordinates?

The equation of a graph in polar coordinates is different from the traditional Cartesian coordinates. It uses the radius (r) and angle (θ) to represent a point on a plane instead of the x and y coordinates. The equation is usually in the form of r = f(θ), where f(θ) is a function of θ.

2. How do you convert an equation in Cartesian coordinates to polar coordinates?

To convert an equation from Cartesian coordinates to polar coordinates, you can use the following formulas:

r = √(x² + y²) - to convert x and y to r

tan θ = y/x - to convert x and y to θ

Once you have the values for r and θ, you can substitute them into the polar equation, r = f(θ), to get the equation in polar coordinates.

3. How do you graph an equation in polar coordinates?

Graphing an equation in polar coordinates involves plotting points on a polar grid using the radius and angle values. For each value of θ, you can calculate the corresponding value of r using the equation. Then, plot the points (r, θ) on the grid and connect them to form the graph.

4. What is the significance of the polar coordinates system?

The polar coordinates system is useful for representing and graphing equations that have circular or symmetrical patterns. It also simplifies some calculations, such as finding the distance between two points, and allows for a different perspective in understanding mathematical concepts.

5. Can you convert a polar equation to Cartesian coordinates?

Yes, you can convert a polar equation to Cartesian coordinates using the following formulas:

x = r*cos(θ) - to convert r and θ to x

y = r*sin(θ) - to convert r and θ to y

Once you have the values for x and y, you can substitute them in the Cartesian equation, y = f(x), to get the equation in Cartesian coordinates.

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