I did make a picture I am confused by the little piece of the cardoid that isn't in the first quadrant.BvU said:
stolencookie said:
As shown in BvU's graph, the region of integration is entirely on the left side of the vertical axis. What is ##\theta## at the upper intersection point? At the lower intersection point? There is also some symmetry you can take advantage of.stolencookie said:
Ambiguous -- in the picture a small piece is missing because I simply didn't grab the full ##\theta## range for the red curvestolencookie said:
A double integral in polar coordinates is a mathematical concept used to find the area of a region on a polar coordinate plane. It involves integrating a function over a two-dimensional region in polar coordinates, which is represented by a double integral symbol.
In a regular double integral, the region of integration is defined by rectangular coordinates (x and y). In contrast, a double integral in polar coordinates uses polar coordinates (r and θ) to define the region of integration, which can be more useful for certain types of problems involving circular or curved regions.
The formula for a double integral in polar coordinates is given by ∬ f(r, θ) r dr dθ, where f(r, θ) is the function being integrated, r is the radius, and θ is the angle in radians. This formula can be used to find the area of a region or to evaluate a double integral over a polar region.
Double integrals in polar coordinates are commonly used in physics and engineering to calculate moments of inertia, center of mass, and other physical quantities for objects with circular or spherical symmetry. They are also used in mathematics to solve problems involving polar curves and surfaces, such as finding the area between two polar curves.
Yes, a double integral in polar coordinates can be converted to a regular double integral by using the Jacobian transformation. This involves changing the limits of integration and replacing the polar variables (r and θ) with their equivalent rectangular coordinates (x and y). However, this process can be more complicated and is not always necessary, as some problems are easier to solve using polar coordinates.
