Chapter 10.5, Problem 52E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Area Sketch the strophoid r = sec θ − 2 cos θ ,   − π 2 < θ < π 2 . Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

To determine

To graph: The strophoid r=secθ2cosθ, π2<θ<π2 and convert this equation into rectangular coordinates and also to find the area by the enclosed loop.

Explanation

Given:

Polar equation is given r=secÎ¸âˆ’2cosÎ¸ and the interval is given âˆ’Ï€2<Î¸<Ï€2

Graph:

Use graphing calculator to sketch graph and area is symmetric about x- axis so the area can be calculated using integration as

A=2âˆ«0Ï€412r2.dÎ¸

Consider, r=secÎ¸âˆ’2cosÎ¸

Graph of curve is shown below:

Now, convert the equation into rectangular coordinates,

Consider, r=secÎ¸âˆ’2cosÎ¸

r=1cosÎ¸âˆ’2cosÎ¸rcosÎ¸=1âˆ’2cos2Î¸rcosÎ¸=1âˆ’2(xr)2

Multiply r2 both sides and get,

r3cosÎ¸=r2âˆ’2r2(xr)2r2(rcosÎ¸)=r2âˆ’2x2(x2+y2)x=x2+y2âˆ’2x2x3+y2x=y2âˆ’x2

That is

x2(1+x)=y2(1âˆ’x)

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