Polar coordinates are a system of representing points in a plane using a distance from a fixed point (the pole) and an angle from a fixed direction (the polar axis).
Polar coordinates are useful for evaluating integrals when the region of integration is better described in terms of a radius and angle rather than x and y coordinates. This is often the case for symmetric shapes such as circles, ellipses, and sectors.
No, not all integrals can be evaluated using polar coordinates. The region of integration must have a defined shape, such as a circle or sector, for polar coordinates to be applicable.
To convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), the following formulas can be used: r = √(x^2 + y^2) and θ = arctan(y/x). Keep in mind that θ is measured counterclockwise from the positive x-axis.
Yes, there are some cases where using polar coordinates can simplify the process of evaluating integrals. This is particularly true for integrals involving symmetric shapes, as mentioned earlier. Additionally, polar coordinates can also be useful for evaluating integrals involving curves that are difficult to express in Cartesian coordinates.