Question

Use one of the formulas below to find the area under one arch of the cycloid x = t − sin(t), y = 1 − cos(t). a = c x dy = − c y dx = 1 2 c x dy − y

Answer 1

(Green's Theorem" The area is bound by the x-axis on the bottom part from x = 0 to x = 2Ď€, and by the cycloid on the top. C = the bounding curve Csub1 = the x-axis part of C Csub2 =the cycloid part. You will take an integeral 2 x the Area will end up being the integral from 2pi to 0 of cos(t)dt with is 6pi So 2 x Area = 6pi so the area = 3pi.

Answer 2

The area under one arch of the cycloid x = t − sin(t), y = 1 − cos(t). is $3\pi$

Explanation:

Green's theorem is a relationship between the two-dimensional vector field line integral over the closed path in the plane and the double integral over the region as it encloses.

A cycloid is the curve that traced by point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. Cycloid is when a circle rolls along a straight line the path.

What is the area under one arch of the cycloid x = t − sin(t), y = 1 − cos(t) ?

The area is bounded by the x-axis on the bottom from $x = 0$ and $x = 2 \pi$ and  by the cycloid on the top. Whereas the C is the bounding curve that consist of  the x-axis traversed from $x = 0$ to $x = 2 \pi$followed by the cycloid going from $t = 2\pi$ to $t = 0$. $C_1$ is the x-axis part of C and $C_2$ is the cycloid part

$2 * area(D) = \int\limits^._C {-y dx+x dy} \, = \int\limits^._{C1} {-y dx+x dy +} \, \int\limits^._{C2} {-y dx+x dy \,$

$2 * area(D) = \int\limits^._{C2} {-y dx+x dy} \,$ (because $y=0$ and $dy=0$ on $C_1$}

Because of x = t − sin(t), y = 1 − cos(t) then

$2 * area(D) = \int\limits^{t=0}_{t=2\pi} {(-(1-cost)(1-cost)+(t-sint)sint)dt} \,$

$2 * area(D) = \int\limits^{t=0}_{t=2\pi} {(-2+2cost+tsint)dt=4\pi-tcost|_{t=2\pi}^{t=0} + \int\limits^{t=0}_{t=2\pi} {cos t} \, dt = 6\pi } \,$

Therefore the area under one arch of the cycloid is $area (D) = 3\pi$

Learn more about one arch of the cycloid  : https://brainly.com/question/10582117 and https://brainly.com/question/14168141

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