 So I mentioned before that calculating surface area When it comes to revolution is very much similar to calculating the volume of a solid revolution And if you remember with the solids revolution, we had the theorem PAP as available to us to that in many cases We could simplify the calculation of the volume if we could identify the centroid of The of the region that is being turned and that's being rotated turned out a similar statement can be said for Surface area if we have a closed curve So it say let's see be a closed curve in the plane that lies entirely on one side of a line L So what we mean by a closed curve is that this curve comes to itself I have some type of blobby thing right here, but it's closed Which means it starts and stops at the same location. And so we have some axis L Like this and we want to rotate this curve around L So think of that surface so what we have here in front of us is a very poorly designed Donuts, but if we have a closed curve and we were to spin this around Some line L the area of the resulting surface is going to be the product of the arc length of C and The distance traveled by the centroid of C. So what we mean by that is if you take the centroid Which might look something like this. It's going to travel some circular distance around this axis right here let's look at the circumference of that thing and then We also want to consider the length of this arc of This arc C here. So you have some arc length s And so what the theorem of PAP is is going to tell us is that the surface area here It's going to equal the circumference of the centroid multiplied by the arc length of The curve and so that's simple product if we can find them can help us out dramatically here So as an example, let's take a donut remember a torus is formed by rotating a circle Of radius little r about a line in the plane Of the circle that is a distance capital R, which should be bigger than little r from this from the center of the circle So if you have some circle Like this and you rotate around some line Like so so this is a circle whose radius is r and then the distance of the center to the axis is capital R If you rotate this thing around the axis you get a so-called torus Which again is just a is a donut like shape object Where in this donut if you take a some type of cross-sectional Slice which is itself a circle it has little radius r and the center of the donut to this to this value here's capital That's what we mean by this this torus here. So what's the surface area of a torus? So we've talked about the volume of the torus before so that's how much dough you need to make a donut The surface area would be like how much glaze or how much frosting do we want to put on the doughnuts a glaze Don't know you typically covered all the way around with glaze Maybe a frosted donut might just have frosting on top So we take half of the surface area This could be a very difficult integral to set up and solve but with the theorem of PAP as it adapted for surface area It actually turns out really nicely the the centroid here as it travels around As it travels around this whole thing the circumference of that thing Well the circumference of that thing as the centroid travels around is going to be 2 pi capital R And then if you take the arc length of this of the curve, which is just the circumference of this little circle that's going to be 2 pi little r and so by the theorem of PAP is the Not the volume here the surface area is going to equal 2 pi r times 2 pi r Those are different r's r which case you get 4 pi squared little r capital r very simple formula to find out how much glaze you need for your donut So when one can use the theorem of PAP as it can dramatically simplify this calculation We're going to love that but one thing we're going to see here is that the the ability to do that First of all requires we can calculate an arc length, right? Now for a circle, it's not so bad, but generally this requires an integral Okay, also what we're going to see in future lectures is that the centroid typically requires an Integral as well now for a circle. It's easy to find But centroids typically require you take an integral to find them and likewise arc length things integrals So in general practice The theorem of PAP is might not be very useful because if it takes an integral to find the centroid an integral to find the arc length Multiplying two integrals together is not easier than finding a single integral So oftentimes the formula we have before works better But if there are non-calculus ways of finding the arc length and non-calculus ways finding the centroid Like it was for this torus then we have a very slick way of finding the surface area using the theorem of PAP is again