 Hello, and welcome to the screencast on Section 11.8, Triple Integrals in Cylindrical and Spherical Coordinates. In this screencast, we will introduce Cylindrical and Spherical Coordinates. We'll learn about triple integrals in these coordinates during the work, and, in fact, let's start with Cylindrical Coordinates, which should look familiar from our work earlier using polar coordinates. The Cylindrical coordinates of Appointment R3 are given by R, Theta, and Z, where R and Theta are the polar coordinates of the point X, Y, and Z is the same Z coordinate as in the Cartesian coordinates. You can see a picture of what this looks like from it. To convert between Cartesian and Cylindrical coordinates, we use the same identity as we developed with polar coordinates. First, with the cylindrical, we see that R equals the square root of X squared plus Y squared. The tangent of Theta equals Y over X. Both of these are the same as we have for polar coordinates. And lastly, we see that Z equals Z, which Z just remains. And this is all assuming that next to convert from Cylindrical to Cartesian, we again see some familiar line of Theta and the spherical coordinates of Appointment Theta and Theta, where R is the distance from the point to the origin. Theta has the same interpretation as it does in polar coordinates. That Theta is the angle the X, Y coordinates makes. So if we take the point P and we drop it down to the X, Y plane, Theta is going to... And lastly, P is the angle between the positive Z axis and the line segment from the origin to the point. Line segment from the origin to this point from this segment, the positive Z axis is here. Let's look at how to convert between Cartesian coordinates and spherical coordinates. So converting from Cartesian to spherical first, we are given the Cartesian coordinates of Appoint P in the spherical coordinates of P satisfied the following three equations. The first is that Rho is equal to the square root of X squared plus Y squared plus P. The second should look familiar from Cylindrical and polar coordinates. That tangent of Theta is... And the third is the expression we'll use to get P. That cosine of P is the latter two equations. We require two things, that X is non-zero. On the other hand, converting from spherical to Cartesian, if we are given the spherical coordinates of Appoint P, then the Cartesian coordinates of P satisfy the following three equations. First, X is equal to Rho times sine of P times cosine of Theta.