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## Calculus 2

### Unit 5: Lesson 7

Area: polar regions (single curve)# Worked example: Area enclosed by cardioid

AP.CALC:

CHA‑5 (EU)

, CHA‑5.D (LO)

, CHA‑5.D.1 (EK)

, CHA‑5.D.2 (EK)

Area enclosed by cardioid.

## Want to join the conversation?

- Is there a video on Cardioids?

I feel like Sal sort of introduced us to this without teaching us about it.(27 votes)- It is surprisingly hard to find videos on these topics! Some of the best ones I've found on YouTube are by Krista King, PatrickJMT, and Blackpenredpen. Hope this helps!(2 votes)

- I don't get how Sal got alpha and beta.

Can someone explain to me how Sal got those numbers please?(7 votes)- alpha and beta are the bounds of the integral. This integral has no visible bounds but the bounds can be set at 360 degrees = 2pi rad and 0 degrees = 0 rad since the shape covers the whole circle.(5 votes)

- When solving for Cardioid problems, are we always going to be presented with a graph, or will there be problems where we are only given the r equation, and we have to draw the graph ourselves or imagine the graph.(4 votes)
- i would imagine that sometimes, you won't be given a graph.(2 votes)

- At5:06, isn't the cos^2 identity 1/2(theta+sin(theta)cos(theta))? Why is it different in this case?(2 votes)
- cos²u = ½(1 + cos(2u))

This is one of several Power Reduction Formulas.

Check out:

http://www.sosmath.com/trig/Trig5/trig5/trig5.html

and

https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Vi.C3.A8te.27s_infinite_product(3 votes)

- How do you know if Beta is 2 pi, 4pi, 6pi, 8pi, 10pi or other multiples of 2pi radians?(2 votes)
- This depends on the specific function, here it makes a full loop at 2pi radians, s if you have beta be greater than 2pi you will be counting the area of a second loop. 4pi would essentially have you take the area of the shape twice, go on and try it.

So the takeaway is to always realize how many radians it takes for a curve to make a full cycle if there is one . If a curve doesn't go back to the start it's a little bit more tricky , but just be aware where it's taking the area of, and that it may double count parts of it.

Let me know if that didn't help(3 votes)

- How is the Beta for the cardioid 2pi?(1 vote)
- We're going from the start of the function(alpha) to the end of it(beta). In this case, the function starts at zero, and ends at 2pi. So, alpha is zero, beta is 2pi.

Hope it helps.(5 votes)

- is there any other way to integrate the r(0)^2 and solve for the area without using the trig identity besides just plugging the whole thing into the math 9 function into the calculator?(2 votes)
- I mistakenly chose pi/6 and 11pi/6 as my alpha and beta respectively. Did anyone else make the same mistake? And how would one know the graph encompassed an entire circle?(1 vote)
- This is definitely a mistake I've made on several problems. I've found that the best way to be sure that your bounds of integration are correct is to set the expression r(𝛉) equal to zero. This will give you candidates for the bounds of integration. Sometimes I like to do this without looking at the graph first because then I don't have any mental picture of what I think the bounds should be that might mess up my calculations. This is also much better than eyeballing the bounds of integration, which is very tempting for curves like these, but often can result in wrong answers due to very small regions where the function crosses a certain axes, or achieves a certain value that is very difficult to see just by eyeballing it. For this graph, if you were to zoom in very closely (try plotting this graph on demos.com if you would like) you would see that the graph just barely grazes the positive x-axis. Because of this, you must use 0 to 2π as your bounds of integration, otherwise you will cut off this tiny tiny piece on the positive x-axis, and your area will be wrong. I hope this helps some! It's definitely an easy mistake to make, so I think the best advice I can give is try to reason through it mathematically, not graphically, as the graph will sometimes trick you. :)(3 votes)

- So will the a and b of integration always be 0 and 2 pi for full cardioids?(2 votes)
- No, it should be done when you set r = 0. In the case of this video, since cos(0) = 1 you subtract 1 - 1 = 0. So the first instant when r = 0 is when cos(theta) = 1. To get the next instant when cos(theta) = 1 is by completing one full rotation (adding 2pi). It doesn't work for every case, but just start by setting r = 0 and finding what you plug in for theta that will make the equation equal to 0 and then finding the next radian that will grant you the same value.(1 vote)

- 5:13seconds. should it be negative - 1/4sin2(theta)? instead of Positive +1/4sin2theta?(1 vote)
- No, its positive. The integral of cos is sin. You may be thinking of the integral of sin which is negative cos. None of the coefficients affect the sign.(2 votes)

## Video transcript

- So this darker curve in blue is the graph of r is equal
to 1 minus cosine of theta, of course we're dealing
in polar coordinates here. And what I'm interested in is
to see if we can figure out the area enclosed by this curve. And I encourage you to pause the video and try it on your own. Alright, let's work through it together. So we've already seen, we've
already given ourselves the intuition for the formula, that the area enclosed by a polar graph is going to be equal to one
half the definite integral from our starting theta
to our ending theta, from alpha to beta, of r
of theta squared d theta. And so we, essentially,
just have to apply this to this function right over here. So in this case, the area
is going to be equal to one half the definite integral. Now what's our alpha and what's our beta? Well, we're going from theta
is equal to zero radians, and we're essentially going all the way-- When theta is equal to
zero radians is 1 minus 1 we're right over there. And then we go all the way around to theta is equal to two pi radians. Notice when we're back at two
pi, cosine of two pi is one, one minus one is zero again. So we get back to that point. So we're going from theta
is equal to zero radians to theta is equal to two pi radians. Now what's r of theta squared? Maybe I'll color code this a little bit. R of theta squared. Well it's just going to be
one minus cosine of theta. One minus cosine theta squared. And of course we have our d theta. We have our d theta. And now we just have to
evaluate this integral. So once again, at any
point you feel inspired, try to evaluate this. So let's do this, alright. So what I would do... So this is going to be equal to one half times the definite integral
from zero to two pi. And let me expand this out. This is going to be one
minus two cosine theta plus cosine squared theta, d theta. D theta. Now I know how to take the
anti-derivative of one, I know how to take the anti-derivitive of negative cosine of theta,
but cosine squared theta, this is a little bit... It doesn't jump out at you
that you can just do this, use u-substitution or something like that, but lucky for us, we have
our trigonometric identities, and so we know. We know that cosine squared of theta is just the same thing
as one half times one plus cosine of two theta. You learned this in trigonometry class. If you didn't, we'll,
you've learned it just now. And that's why this is
one of the more useful trigonometric identities if
you're finding any type of anti-derivative or if
you're integrating anything. And so let's do that. Let's rewrite this right over here as one half times one plus
cosine of two theta. And let's see, and maybe we could... Yeah, let's just do it like that. I guess we could, if we wanted... Well, we'll just do it like that. So this is going to be equal to one half, and then we are going to... One half, now let's just
start taking anti-derivatives. One half. Now the anti-derivative of
one, with respect to theta, is just going to be theta. The anti-derivative of
negative two cosine of theta, well that's just going to be
negative two sine of theta. Negative two sine theta. You can take the derivative, the derivative of sine is cosine and the negative two,
it'll just multiple it times the derivative of sine of theta so it's negative two cosine of theta. And then we're going to have, let's see... So let me distribute this. This is the same thing
as one half plus one half cosine of two theta. So let's just assume it's this way. So the anti-derivative of one half, so the anti-derivative of one-half. So I'm really looking at
that right over there. It's going to be one half theta. One half theta. And then the anti-derivative of one half cosine of two theta, let's see... The derivative of sine of two theta is two cosine of two theta. So the anti-derivative of this is... The anti-derivative of
cosine of two theta, and you can do u-substitution if you like, but you might be able
to do this in your head, the anti-derivative of cosine
of two theta is going to be one half sine of two theta. And then you have this
one half right over here. So this is going to be... Let me show you what I'm
finding the anti-derivative of. Of that right over there, of this, and I guess this right over here. So this is going to be plus
one fourth sine of two theta. And I encourage you to
find the derivative here if that last part was
a little bit confusing. The derivative of sine of two theta is two cosine of two theta, two over one fourth is one half, you get to one half cosine of two theta. And we're going to
evaluate that at two pi, at two pi, and at zero. So when you evaluate
it, one thing that might jump out at you is when
you evaluate this at zero, this whole thing,
everything, every term here is just going to be zero, so
that simplifies things nicely. So we really just have
to take one half of... It evaluated at two pi. So this is going to be
one half times two pi, two pi, and then sine of two pi is zero, so that's just going to be zero. And then plus one half times two pi, so that's going to be plus pi. And then sign of two times
two pi sine of four pi, that's still going to be zero. So this is going to be zero as well, and we are almost done. So this is going to be
one half times three pi or three halves, three
halves pi is the area, is the area of this region.