Ejemplo 01 - Ingtegrales polares

Q & A

What is the main concept introduced in this script?

The main concept is how to solve double integrals in polar coordinates, especially for circular regions, and how to interpret the results geometrically.

What does the double integral in polar coordinates look like?

In polar coordinates, the double integral is given by a function in terms of theta, and the area differential is expressed as r * dr * dtheta.

What is the key feature of the region 'r' in the example?

The region 'r' is a disk centered at the origin with a radius of 5, which is important for determining the limits of the integral.

Why does the function x get replaced by r cos(θ) in the integral?

In polar coordinates, the Cartesian coordinates (x, y) are replaced by polar coordinates, and x is represented as r * cos(θ). This substitution is part of the conversion from rectangular to polar coordinates.

What are the limits of integration for 'r' and 'θ'?

The limits for 'r' are from 0 to 5, corresponding to the radius of the disk. The limits for θ are from 0 to 2π, representing a full rotation around the circle.

How does the integral become easier to solve with polar coordinates?

In polar coordinates, constants like cos(θ) can be factored out of the integral, simplifying the process. The integration then involves simpler terms, like r², making it easier to solve.

What is the result of the inner integral with respect to 'r'?

The inner integral with respect to 'r' results in r³ / 3, evaluated from 0 to 5, which gives the value of 125/3.

What is the result of the outer integral with respect to 'θ'?

The outer integral with respect to θ results in the integral of cos(θ), which gives sin(θ) evaluated from 0 to 2π. Both sin(2π) and sin(0) are 0, so the integral results in 0.

Why does the final result of the integral equal 0?

The final result is 0 because the integral represents the volume of a solid, and the positive and negative parts of the volume cancel each other out. The solid is a cylinder cut by the plane z = x, with an equal volume above and below the x-y plane.

How can we visualize the solid formed by the integral?

The solid is a cylinder with a circular base. The function z = x represents a plane that cuts through this cylinder, forming a symmetric volume above and below the x-y plane, which leads to the final integral result of zero.