7.2 Calculus of Parametric Curves

[STARTS AT 11:58]

Instructor: So let's go and find the area under the curve of the cycloid defined by these equations. 

So our area is going to be equal to the integral from 0 to 2 pi of y. That's 1 minus cosine t times x prime, which is 1 minus cosine t dt. All right. 

First I'm going to multiply this out. That'll be 1 minus 2 cosine t plus cosine squared t dt. Now going back to our section on trigonometric integrals, we have this identity where cosine squared t is 1 plus-- or I'll write it differently. It is one half plus cosine 2t over 2. Bring down the rest of this. One dt integral from 0 to 2 pi. Combining like terms there, you have 3/2 minus 2 cosine t plus 1/2-- or I'll leave that over 4. Cosine 2t over t for 2 dt. 

OK, now we can take the antiderivative. 3 over 2t minus 2 sine t plus sine 2t over 4, evaluated from 0 to 2 pi. And I'll save you that work there. For that we are going to get 3 pi. So the area underneath that cycloid is 3 pi.

[ENDS AT 14:15]