Double Integrals (polar coordinate)

In summary, a double integral in polar coordinates is a mathematical concept used to calculate the area under a curve on a two-dimensional polar coordinate plane. It differs from a regular double integral in that the boundaries of integration are defined by polar coordinates instead of rectangular coordinates. The formula for calculating a double integral in polar coordinates is ∫∫f(r,θ) rdrdθ, and it can be converted from rectangular coordinates using substitutions and adjusting the limits of integration. Real-world applications of double integrals in polar coordinates include calculating physical properties and potentials of objects with circular symmetry in fields such as physics and engineering.
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θ goes from 0 to π/2 to cover the first quadrant.
 

FAQ: Double Integrals (polar coordinate)

1. What is a double integral in polar coordinates?

A double integral in polar coordinates is a mathematical concept used to calculate the area under a curve on a two-dimensional polar coordinate plane. It involves integrating a function with respect to both the radial distance and the angular direction.

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

In a regular double integral, the boundaries of integration are defined by rectangular coordinates (x and y). In a double integral in polar coordinates, the boundaries are defined by the polar coordinates (r and θ).

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

The formula for calculating a double integral in polar coordinates is ∫∫f(r,θ) rdrdθ, where the limits of integration for r and θ are determined by the shape and size of the region being integrated.

4. How do I convert a double integral in rectangular coordinates to polar coordinates?

To convert a double integral from rectangular coordinates to polar coordinates, you can use the substitution r = √(x^2 + y^2) and θ = tan^-1(y/x). Then, you can rewrite the function being integrated in terms of r and θ and change the limits of integration accordingly.

5. What are some real-world applications of double integrals in polar coordinates?

Double integrals in polar coordinates are commonly used in physics, engineering, and other scientific fields to calculate the moments of inertia, center of mass, and other physical properties of objects with circular symmetry. They are also used in calculating the electric and gravitational potential of objects with a circular distribution of charge or mass.

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