   Chapter 10, Problem 2P

Chapter
Section
Textbook Problem

(a) Find the highest and lowest points on the curve x4 + y4 = x2 + y2. (b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines y = ±x, so it suffices to consider y ≥ x ≥ 0 initially.) (c) Use polar coordinates and a computer algebra system to find the area enclosed by the curve.

(a)

To determine

To find: The highest and lowest point on the curve.

Explanation

Given:

The given curve is as below.

x4+y4=x2+y2 (1)

Calculation:

Differentiate the equation (1) with respect to x.

x4+y4=x2+y2(4x3)+(4y3.y')=(2x)+(2y.y')4x32x+4y3y'2y.y'=04x32x+y'(4y32y)=0y'(4y32y)=4x3+2xy'(4y32y)=2x(12x2)

y'=2x(12x2)(4y32y)y'=2x(12x2)2y(2y21)y'=x(12x2)y(2y21)

Take the value of x at which the condition y'=0 as true.

When the condition x=0 is true, the value of y' is as below.

y'=x(12x2)y(2y21)=0(12(02))y(2y21)=0

When the condition x=12 is true, the value of y' is as below.

y'=x(12x2)y(2y21)=12(12(12)2)y(2y21)=0

When the condition x=12 is true, the value of y' is as below.

y'=x(12x2)y(2y21)=12(12(12)2)y(2y21)=0

Therefore, the condition y'=0 is true when the value of x=0 and x=±12

(b)

To determine

To sketch: The curve.

(c)

To determine

To find: The area enclosed by the curve.

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