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.


I just want to review the COVID-19 numbers for Monday the 17th. Compared to last week, we're up to 887 positive cases. That's up 83 from last week, which is a slight decrease from the percentage or the number that we've had the previous two weeks. We have 105 active cases, and that's down 40 active cases from 145 last week. We have 774 cases that have recovered, and that's up 123 from last week. We had three people in the hospital last week, and now we have seven, so that's an increase of four. We're still standing with eight people that have passed away due to the COVID virus. And we have 24,733 negative tests, and that's up 2,172 from last week. And the percentage of positive tests to the total test is standing at 3.4%, and that's remained the same from last week. The COVID-19 is a terrible virus that we continue to learn more about every day. However, we do know some of the best practices, including masks and face cloth coverings, physical distance, and keeping our social circles small. These are all key to slowing the spread of the virus. We also know that working together is a key to our lives, returning to normal. A new ordinance is going to be introduced on Tuesday, August 18th, tomorrow at the Sheboygan County Board of Supervisors meeting regarding the coronavirus pandemic. This ordinance provides for enforcement of public health orders and legislators oversight. At this meeting, the ordinance will then be referred to a committee for a recommendation to return to the county board for a final vote on September 15th. The goal of the proposed ordinance is to be prepared, to be used, if we see a significant spike in the COVID-19 insurgents in our hospitals, and if they are on the brink of being swamped with cases. And then the county public health officer does have significant authority in state law and statutes, and is proposed that this ordinance will soften the existing authority as it proposes checks and balances with legislated oversight by the county board. There are currently no local public health orders in place. If we do see a surge of COVID-19 in our community and need to take stronger action to protect the health and vitality of our community, we will have a stronger collaborative approach established and in place. The purpose of this ordinance is to be prepared if the situation gets significantly worse, and if the greater countywide action needs to be taken. The ordinance also includes an enforcement component that is easier to administrate and less punitive and again only if needed. This week you will see new billboards and a bus wrap ads to ask everyone to do their part to stop the spread of the coronavirus in Sheboygan County. Please wear a mask and maintain social distance. These are challenging and unprecedented times and will take all of us working together to keep our community safe and our businesses open.