Double Integration with Polar Coordinates

In summary, a double integral with polar is a type of integral used to calculate the area between two curves in polar coordinates. It differs from a regular double integral as it uses polar coordinates and has a specific formula for calculation. Applications of this type of integral include calculating moments of inertia, areas and volumes of complex shapes, electric and gravitational fields, and solving problems in fluid mechanics. Some tips for solving double integrals with polar include sketching the region of integration, expressing the integrand in polar coordinates, choosing the order of integration, manipulating integration limits, using symmetry, and checking the answer using a graphing calculator or software.
  • #1
Renzokuken
14
0
1. Integrate f(x,y)=x+y
1<=x^2+y^2<=4, x>=0, y>=0


3. ∬x+y dxdy x=rcos(o) y=rsin(o)

∬r(rcos(o)+rsin(o))drdo
r is from 1 to 4, o is from 0 to pi/2


I get the wrong answer and don't know why
 
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  • #2
[tex]x^2+y^2=r^2[/tex]

r doesn't go from 1 to 4. r^2 goes from 1 to 4.
the 0 to pi/2 seems good.

Don't know about the rest of the integral though sorry haven't learned polars yet :/
 

1. What is a double integral with polar?

A double integral with polar is a type of integral that is used to calculate the area between two curves in polar coordinates. It involves integrating a function over a region in the polar plane, which is defined by two polar curves.

2. How is a double integral with polar different from a regular double integral?

A double integral with polar is different from a regular double integral because it uses polar coordinates instead of rectangular coordinates. This means that the integration limits and the integrand are expressed in terms of polar variables, such as r and θ, instead of x and y.

3. What is the formula for calculating a double integral with polar?

The formula for calculating a double integral with polar is:
R f(r, θ) dA = ∫θ=abr=r1(θ)r2(θ) f(r, θ) r dr dθ
where R is the region of integration, a and b are the limits of integration for θ, and r1(θ) and r2(θ) are the polar curves that define the region of integration.

4. What are some applications of double integrals with polar?

Double integrals with polar are commonly used in physics and engineering to calculate the moments of inertia of objects, as well as the areas and volumes of complex shapes. They are also used in calculating electric and gravitational fields, and in solving problems in fluid mechanics.

5. What are some tips for solving double integrals with polar?

Some tips for solving double integrals with polar include:
- Sketching the region of integration and identifying the polar curves that define it
- Expressing the integrand in terms of polar coordinates
- Choosing the order of integration based on the shape of the region
- Manipulating the limits of integration to make the integration process easier
- Using symmetry to simplify the integration process
- Checking your answer by converting the polar integral to a rectangular integral and evaluating it using a graphing calculator or software.

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