Steff_Rees said:Homework Statement
[tex]\int_{y=-\infty}^{\infty} \int_{x=-\infty}^{\infty} \frac{x^4+y^4}{(1+x^2+y^2)^4} dx dy[/tex]
Homework Equations
i'm not sure what the new limits are after the transformation to polar coordinates and how to solve the integral.
The Attempt at a Solution
i have my new function of integration to be
[tex] \frac{r^4 cos^4(\theta)+r^4 sin^4(\theta)}{(1+r^2)^4} r dr d\theta[/tex]
Polar coordinates are a system of coordinates used to describe the position of a point in a two-dimensional space. They use two values, the distance from the origin (r) and the angle from a fixed reference line (θ), to specify the location of a point.
A double integral in polar coordinates is a mathematical concept used to find the area under a curve in a polar coordinate system. It involves integrating a function over a region in the polar plane, using the distance from the origin and the angle as the variables of integration.
To convert a double integral from Cartesian to polar coordinates, you need to substitute the Cartesian variables (x and y) with their corresponding polar expressions (r cos θ and r sin θ) and use the Jacobian of the transformation (r) to change the limits of integration.
Double integrals in polar coordinates have various applications in physics, engineering, and economics. They are used to calculate moments of inertia, to find the center of mass of a two-dimensional object, and to solve problems involving circular symmetry, such as electric fields and fluid flow.
Some common mistakes when solving a double integral in polar coordinates include forgetting to convert the limits of integration, using the wrong Jacobian, and forgetting to include the r term when integrating. It is also important to check for symmetry and use the correct expression for the function being integrated.