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Calculus 2

Course: Calculus 2>Unit 5

Lesson 7: Area: polar regions (single curve)

Worked example: Area enclosed by cardioid

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.
• 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!
• I don't get how Sal got alpha and beta.
Can someone explain to me how Sal got those numbers please?
• 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.
• 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.
• i would imagine that sometimes, you won't be given a graph.
• How do you know if Beta is 2 pi, 4pi, 6pi, 8pi, 10pi or other multiples of 2pi radians?
• 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
• How is the Beta for the cardioid 2pi?
• 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.
• ok, is trig identities important? do i have to memorize it
• Yes, they are important. Depending on the course they may allow a formula sheet with trig identities but in general its always good to memorize atleast a few important ones or know how to derive them on the spot. cos(x)^2 = (1/2)(1+cos(2x)) is pretty useful to memorize and for taking antiderivatives
(1 vote)
• At , isn't the cos^2 identity 1/2(theta+sin(theta)cos(theta))? Why is it different in this case?
• How do you know which trig identity useful? I don't want to memorize them, but I do want to know when and which to re-derive them from simpler identities.