   Chapter 16.4, Problem 19E

Chapter
Section
Textbook Problem

Use one of the formulas in (5) to find the area under one arch of the cycloid x = t − sin t, y = 1 − cos t.

To determine

To find: The area under one arch of the cycloid.

Explanation

Given data:

Parametric equations are x=tsint and y=1cost .

Consider a curve C1 , 0t2π as arch of the cycloid from (0,0) to (2π,0) with parametric equations

x=tsint (1)

y=1cost (2)

Differentiate equation (1) with respect to t.

ddt(x)=ddt(tsint)dxdt=ddt(t)ddt(sint)dxdt=1cost {ddt(t)=1,ddt(sint)=cost}dx=(1cost)dt

Differentiate equation (2) with respect to t.

ddt(y)=ddt(1cost)dydt=ddt(1)ddt(cost)dydt=0(sint){ddt(k)=0,ddt(cost)=sint}dy=sintdt

Consider a curve C2 , 0t2π as a segment from (2π,0) to (0,0) with parametric equations

x=2πt (3)

y=0 (4)

Differentiate equation (3) with respect to t.

ddt(x)=ddt(2πt)dxdt=ddt(2π)ddt(t)dxdt=01 {ddt(t)=1,ddt(k)=0}dx=dt

Differentiate equation (4) with respect to t.

ddt(y)=ddt(0)dydt=0 {ddt(0)=0}dy=0dt

The area of curve C, which is one arch of cycloid is the union of regions of curves C1 and C2

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