Area & Arc Length of a Cycloid (one arch)

Which one did you like more?
Area of a cycloid, 1:00
Arc Length of a Cycloid, 8:45

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blackpenredpen | 曹老師

[letsseepaulallenscard.]

Comfy bprp maths after a long day of work is best maths

[Cosine_Wave]

0:45 The arc length and area can be calculated by using y = 2r(arccos(x/r) + sqrt(1-(x/r)^2)) and integrating between -r and r. It's not exactly what you asked for, but it will give the correct answer. The difference is that the circle is rolling on the y axis instead of the x (along with a few simplifications to the equation). I also don't know how hard that is to integrate.

[XZellTheBest]

I prefer the first integral. I'm not a math student, I'm just an agronomist, but I love your videos. Please keep going! Greetings from Italy!

[JonahFoley]

10:44 the true fans watch the whole video :)

[StuartSimon]

The cycloid passes the vertical line test, so it is actually theoretically the graph of some function f. I was thinking that you could come up with a closed form of f(x) for the first arch by defining a function g on the interval [0, 2pi). Then you could define f as f(x) = g(x mod 2pi).

[sander_bouwhuis]

I love your enthusiasm! Thanks for working this out very clearly.

[MrJapogm]

10:43 Watching the whole video. Awesome curve. Mind-blowing properties.

[dsantistevan]

I remember doing this and the rotation volume of a cycloid for high school, so cool. Nice vid

[rollinginthedeep]

10:43 hahah, thank you for being an awesome tutor. I like watching your explanations because they're clear and you remind me that math should be fun! It's hard to remember to enjoy learning when exam season comes around, so thank you for all your videos and your help :)

[Linuxmusica]

The area is 3 times the area of the circumference with r radius.

[yurusan]

Although I didn't take calculus in high school, I still like your videos.
They are so satisfying to watch. 10:45

[WandiiNgwenya]

10:44 true fan right here. All the way from South Africa❤‍🔥❤‍🔥❤‍🔥❤‍🔥

[frozenmoon]

1+1 = 2 - you've got to say it with *authority*. I never have been told before in my life with such enthusiasm that 1+1 = 2.

[EpochIsEpic]

10:44 Still watching. I’m gunna have to stop so I still have time to do my precalc HW. Looking forward to tomorrow’s video though!

[carissagamboa]

Why do you make this sound so exciting ! Lol I am having a hard time, but your videos are so encouraging! Thank you!!

[EpochIsEpic]

Well, this went WAY over my head but still interesting to watch 😂

[saw]

At first glance the integral for the arc length looked more difficult. The area integral was simple if you knew the identity. Then again you make everything look easy hahaha

[igorjasenovski]

Pretty good video. Thank you for your beautiful job. Keep it up.

[peterchan]

Cartesian equation of the cycloid
x = r arccos(1 - y/r) - √[y(2r - y)]

[peterchan]

First thing first . . . There was no mention whatsoever on HOW the parametric equation of the cycloid was derived.
And by the way there indeed IS a Cartesian way of writing the equation (without using parameters).

Thus the equation of the cycloid as the locus of a point on the circumference of a circle radius r rolling on the x-axis, beginning from the origin, is given by . . .

x = arccos(1 - y/r)) - sqrt[y(2r - y)]