 Hello and welcome to this screencast on section 11.5, Double Integrals and Polar Coordinates. The rectangular coordinate system is best suited for graphs and regions that are considered over a rectangular grid. The polar coordinate system is an alternative that is a good option for functions and domains that have more circular characteristics. A point P in rectangular coordinates is described by an ordered pair xy, where x is the displacement from P to the y-axis, and y is the displacement from P to the x-axis. A point P can be described with polar coordinates r and theta, where r is the distance from P to the origin, and theta is the angle formed by the line segment from the origin to P in the positive x-axis. Let's look at a picture of what this looks like. Here is the point P. We can see that P can be described by an x-coordinate, which is the displacement from the y-axis, right, how far P is from there, and y-coordinate, which is displacement from the x-axis. That same point P can be described by r and theta, where r is the length of the line segment from the origin to P. Theta is the angle that this line segment makes with the positive x-axis. To convert from rectangular to polar coordinates, we use the Pythagorean theorem and trigonometry. So first, we see that using the Pythagorean theorem, the value of r is the length of the hypotenuse of the right triangle formed here. So r is going to equal the square root of x squared plus y squared. Next, we see again, using right triangle trigonometry, using the fact that tangent of an angle in a right triangle gives us the ratio of the opposite side over the adjacent side, we see that tangent of theta is going to be y over x. This is assuming that x is not equal to zero. There are two important things to note when using this identity to get theta. First, the sign of tangent of theta, whether it's positive or negative, does not uniquely determine the quadrant in which theta lies. So we have to determine the value of theta from the location of the point in the plane. The other thing to note is that the angle theta is not unique, since we could replace theta with a new angle, theta plus 2 pi, and we would still be at the same point. On the other hand, to convert from polar to rectangular coordinates, we use the following two identities. x is going to be equal to r cosine theta and y is going to be r sine of theta.