Find the area bounded by one arc of the cycloid x = a(0-sin(0)), y = a(1-cos(0)),where a > 0, and 0 <= 0 <= 2pi, and the axis(use Green's theorem).

Khadija Wells 2021-01-31 Answered
Find the area bounded by one arc of the cycloid x=a(0sin(0)),y=a(1cos(0)),where a > 0, and 002π, and the axis(use Green's theorem).
You can still ask an expert for help

Want to know more about Green's, Stokes', and the divergence theorem?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

escumantsu
Answered 2021-02-01 Author has 98 answers

Step 1
Note that the atea of the region D bounded by C del D is A(D)=12Cxdyydx.
The given cycloid and the x-axis intersect each other at the points (0,0) and (2 a π,0).
Note that, 12Cxdyydx=12y1+xdyydx+12y2+xdyydx.
The parametrization of the segment y1is{x(t)=ty(t)=0over 0t2aπ.
Step 2
The derivative of the above function is computed as follows,
dxdt=1
dydt=0
Now,
12y1+xdyydx=1202aπ(xdydtydxdt)dt
=1202aπ(1(0)(0)(1))dt
=0
Step 3
The parametrization of the segment y1is{x(0)=a(0sin0)y(0)=a(1cos0)over 002π.
The derivative of the above function is computed as follows,
dxd0=a(1cos0)
dyd0=asin0
Now,
12y2+xdyydx=1202π(xdyd0y
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more