   Chapter 16.4, Problem 20E

Chapter
Section
Textbook Problem

If a circle C with radius 1 rolls along the outside of the circle x2 + y2 = 16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x = 5 cos t − cos 5t, y = 5 sin t − sin 5t. Graph the epicycloid and use (5) to find the area it encloses.

To determine

To find: The graph of epicycloid and the area it encloses.

Explanation

Given data:

Parametric equations of epicycloid are x=5costcos5t , y=5sintsin5t and a circle x2+y2=16 .

Formula used:

Write the area of D using Green’s Theorem (A) .

A=Cxdy (1)

Draw the epicycloid as shown in Figure 1.

Consider the domain of epicycloid as 0t2π .

Write the parametric equation of epicycloids.

x=5costcos5t

y=5sintsin5t (2)

Differentiate equation (2) with respect to t.

ddt(y)=ddt(5sintsin5t)dydt=ddt(5sint)ddt(sin5t)dydt=5cost5cos5t{ddt(sinat)=acos5t}dy=(5cost5cos5t)dt

Substitute 5costcos5t for x and (5cost5cos5t)dt for dy in equation (1),

A=02π(5costcos5t)(5cost5cos5t)dt=02π(25cos2t+5cos25t30costcos5t)dt=[02π(25(12(cos2t+1))+5(12(cos1

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