Question

Example 4.74. Trace the curve x3 + y3 = 3axy
3a sin cos 0
sin + cos e​

Answer 1

Answer:

Answer

Changing to polar form (by putting x=rcosθ,y=rsinθ)

⇒(rcosθ)

3

+(rcosθ)

3

=3arcosθrsinθ

⇒r

3

cos

3

θ+r

3

cos

3

θ=3ar

2

cosθsinθ

⇒r

3

(cos

3

θ+sin

3

θ)=3ar

2

cosθsinθ

⇒r=

(cos

3

θ+sin

3

θ)

3acosθsinθ

Put r=0,sinθcosθ=0

∴θ=0,

2

π

, which are the limits of integration for its loop.

∴ Area of the loop=

2

1

∫

0

2

π

r

2

dθ

=

2

1

∫

0

2

π

(cos

3

θ+sin

3

θ)

9a

2

sinθcosθ

dθ

Divide numerator and denominator by cos

6

θ we get

=

2

9a

2

∫

0

2

π

(1+tan

3

θ)

2

tan

2

θsec

2

θ

dθ

Put 1+tan

3

θ=t and 3tan

2

θsec

2

θdθ=dt we get

=

2

3a

2

∫

1

∞

t

2

dt

=

2

3a

2

∣

∣

∣

∣

∣

−1

t

−1

∣

∣

∣

∣

∣

1

∞

=

2

3a

2

(−0+1)

=

2

3a

2