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## Finding the area of a polar region or the area bounded by a single polar curve

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# Area bounded by polar curves

AP Calc: CHA‑5 (EU), CHA‑5.D (LO), CHA‑5.D.1 (EK), CHA‑5.D.2 (EK)

## Video transcript

- [Voiceover] We now
have a lot of experience finding the areas under
curves when we're dealing with things in rectangular coordinates. So 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 were able to find the area. But now let's move on
to polar coordinates. And in 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. And what I wanna 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 f of theta. 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 it 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 sectors or I guess we could
say little pie pieces? Someone is doing some
serious drilling downstairs. I don't 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 pie pieces and then take the limit as if we had an infinite number of pie pieces? So we want to find the
area of each of these pie pieces and then take the
limit as the pie pieces I guess you could say
become infinitely thin and we have an infinite number of them. And I'll give you one more
hint, for thinking about the area of these pie, I guess you could say the area of these pie wedges. I'll give you another
hint, so if I have a circle I'll do 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 is
obviously more important. All right so if I have
a circle, that's my best attempt at a circle, and it's of radius r and let me draw a sector of this 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 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, 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, what's 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? Well it's going to be a
fraction of the circle. If this is pi, sorry if this
is theta, if we went two pi radians that would be the
whole circle so this is going to be theta over
two pi of the circle. So times theta over two pi would be the area of this sector right over here. Area of the whole circle
times the proprotion of the circle that we've kind of defined or that the sector is made up of. And so this would give
us, the pis cancel out, it would give us one half
r squared times theta. Now what happens if instead of theta, so let's look at each of these over here. So each of these things that I've drawn, let's focus on just one of these wedges. I will highlight it in orange. So instead of the angle
being theta let's just assume it's a really,
really, really small angle. We'll use a differential
although this is a 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 eventually take the limit as our delta
theta approaches zero. But just for conceptual
purposes when we have a infinitely small or super
small change in theta, so let's call that d theta,
and 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 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 instead of my angle being theta I'm calling my angle d theta, this
little differential. So instead of one half
r squared it's going to be, let me do that in a color you can see. This area is going to be
one half r squared d theta. Notice here the angle
was theta, here the angle was d theta, super, super small angle. Now if I wanted to take
the sum of all of these from theta is equal to alpha
to theta is equal to beta and literally there is an
infinite number of these. This is an infinitely small angle. Well then for the entire
area right over here I could just integrate all of these. So that's going to be the
integral from alpha to beta of 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 one half r of
theta squared d theta. Just to remind ourselves or assuming r is a function of theta in this case.