Transform to polar coordinates

In summary, the double integral can be converted to polar coordinates by setting x=1+rcos(θ) and y=1+rsin(θ) and integrating with r ranging from 0 to 1 and θ ranging from -π/2 to 0. The area is a triangle for π/4 < θ < π/2 and for 0 < θ < π/4, the boundary can be written as an equation in r and θ where each line of constant θ has a value of x at the boundary of y=x².
  • #1
dimi212121
2
0
Could someone please convert this double integral to polar coordinates?
0<x<1, x*2<y<1 Int.Int f(x,y)dxdy
 
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  • #2
What is x*2?
 
  • #3
Sorry x^2
 
  • #4
If you draw your area, you ll see that it looks like 1/4 of a circle. Use something like that:
x=1+rcos(fi), y=1+rsin(fi). R will be going from 0 to 1, fi from -pi/2 to 0 then. dxdy gets changed to rdrdfi.
This is not the only solution of course.
 
  • #5
dimi212121 said:
Could someone please convert this double integral to polar coordinates?
0<x<1, x*2<y<1 Int.Int f(x,y)dxdy

(have an int: ∫ and a squared: ² and a theta: θ and a pi: π :smile:)

The area is a triangle for π/4 < θ < π/2.

For 0 < θ < π/4, you simply need to write the boundary as an equation in r and θ.

Hint: for 0 < θ < π/4, each line of constant θ has tanθ = y/x, and it hits the boundary at y = x².

So what is the value of x at the boundary … and what value of r does that correspond to? :smile:
 

1. What are polar coordinates?

Polar coordinates are a type of coordinate system that uses a distance and angle to locate a point in a two-dimensional plane. The distance is measured from the origin and the angle is measured from a fixed reference direction, usually the positive x-axis.

2. How do you convert from rectangular coordinates to polar coordinates?

To convert from rectangular coordinates (x,y) to polar coordinates (r,θ), you can use the following formulas:
r = √(x² + y²)
θ = tan⁻¹ (y/x)

3. What is the purpose of using polar coordinates?

Polar coordinates are often used in situations where it is more convenient to describe a point in terms of its distance and angle from a fixed point, rather than its x and y coordinates. They are also useful for visualizing and analyzing circular or radial patterns in data.

4. How do you convert from polar coordinates to rectangular coordinates?

To convert from polar coordinates (r,θ) to rectangular coordinates (x,y), you can use the following formulas:
x = r cos(θ)
y = r sin(θ)

5. Can polar coordinates be used in three-dimensional space?

While polar coordinates are primarily used in two-dimensional space, they can also be extended to three-dimensional space by adding a third coordinate, typically represented as z. This is known as cylindrical coordinates, and it uses the same principles as polar coordinates but adds a third dimension.

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