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Main content
Current time:0:00Total duration:6:31
AP.CALC:
CHA‑5 (EU)
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Video transcript

we now have a lot of experience finding the areas under curves when we're dealing with things in rectangular coordinates that we saw we took the Riemann sums a bunch of rectangles we took the limit as we had an infinite number of infinitely thin rectangles and we're able to find the area but now let's move on to polar coordinates and polar coordinates I won't say we're finding the area under a curve but really in this example right over here we have a part of the graph of R is equal to f of theta and we've graphed it between theta is equal to alpha and theta is equal to beta what I want to do in this video is come up with a general expression for this area in blue this area that is bounded I guess you could say by those angles and the graph of R is equal to R is equal to f of beta and I want you to come up or at least attempt to come up with an expression on your own but I'll give you a little bit of a hint here when we did in rectangular coordinates we divided things into rectangles over here rectangles don't seem as obvious because they're all kind of coming to this point but what if we could divide things into if we could divide things into sector I guess we could say little pi pieces someone's doing some serious drilling downstairs I don't know if it's picking up on the microphone but anyway I will continue so what would happen if we could divide this into a whole series of kind of PI pieces if we could divide into a whole series of PI pieces and then take the limit as we had an infinite number at the infinite number of pi pieces so where we want to find the area of each of these pi pieces and then take the limit as the PI pieces I guess you could say become infinitely thin and we have an infinite and we have an infinite number of them and I'll give you one more hint I'll give you one more hint for thinking about the area of these pi I guess you could say pi the area of these PI wedges I'll give you another hint so if I have a circle I'm doing my best attempt at a circle luckily the plumbing or whatever is going on downstairs has stopped for now allowing me to focus more on the calculus which obviously more important all right so if I have a circle that's my best attempt at a circle and it's a radius R it's a radius R and then let's say let me draw a sector of the circle it's a sector of a circle so that's obviously R as well and if this angle right here is Theta what is going to be the area what is going to be the area of this sector right over here so that's my hint for you think about what this area is going to be and we're assuming theta is in radians think about what this area is going to be and then see if you can extend that to what we're trying to do here to figure out a somehow I'm giving you a hint again using integration finding an expression for this area so I'm assuming you've had a go at it so first let's think about this so what's the area of the entire the area of the entire circle well we already know that that's going to be PI R squared formula for the area of a circle and then what's going to be the area of this what's going to be a fraction of the circle if this is pi I'm sorry if this is Theta where they if we went two pi radians that would be the whole circle so this is going to be pi this is going to be theta over two pi of the circle so times theta over over 2 pi times theta over 2 pi would be the area of this sector right over here area of the whole circle times the proportion of the circle that we've kind of we have defined or that the sector is made up of and so this would give us this would give us the PI's cancel out it would give us 1/2 R squared times theta now what happens if instead of theta so let's let's look at each of these over here so each of these things that I've drawn so let's focus just on one of these one of these wedges I will highlight it in orange I'll highlight it in orange so instead of the angle being dead let's just assume it's a really really really small angle we'll use a differential all this is a little bit of loosey-goosey mathematics but the important here is to give you the conceptual understanding I could call it a delta theta and then to eventually take the limit as our delta theta approaches zero but I'm just going to just for conceptual purposes when we have a infinitely small or super small change in theta so let's call that D theta and D theta and this this the the radius here or I guess we could say this length right over here you could view it as the radius of at least the arc right at that point it's going to be R it's going to be R as a function of the Thetas that we're around right over here but we're just going to call that our R right over there and so what is going to be the area of this little sector well the area of this little sector is going to set up my angle being theta I'm calling my angle D theta this little differential so instead of 1/2 R squared it's going to be let me do that in a color you can see this this is this area it's going to be 1/2 R squared D theta notice here the angle will stata here the angle is just D theta super super small angle now if I wanted to take the sum of all of these from theta is equal to alpha 2 theta is equal to beta and literally this is an infinite number of these is an infinitely small angle well then for the entire for the entire area right over here I could just integrate I could just integrate all of these so that's going to be the integral from alpha to beta of 1/2 R squared one-half R squared D theta where R of course is a function of theta so you could even write it this way you could write it as the integral from alpha to beta of 1/2 R of theta squared D theta just to remind ourselves we're assuming R is a function of theta in this case