Chapter 16.4, Problem 20E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If a circle *C* with radius 1 rolls along the outside of the circle *x*^{2} + *y*^{2} = 16, a fixed point *P* on *C* traces out a curve called an *epicycloid*, with parametric equations *x* = 5 cos *t* − cos 5*t*, *y* = 5 sin *t* − sin 5*t*. 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

**Formula used:**

Write the area of D using Greenâ€™s Theorem

Draw the epicycloid as shown in Figure 1.

Consider the domain of epicycloid as

Write the parametric equation of epicycloids.

Differentiate equation (2) with respect to *t*.

Substitute
*x* and

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