1000 Solved Problems in Modern Physics

(Tina Meador) #1

54 1 Mathematical Physics


1.48x^2 /^3 +y^2 /^3 =a^2 /^3 (1)


The arcABgenerates only one half of the surface.

Sx
2

= 2 π

∫b

a

y

[

1 +

(

dy
dx

) 2 ]^1 /^2

dx (2)

From (1) we find
dy
dx

=−

y^1 /^3
x^1 /^3

;y=

(

a

(^23)
−x


23 )^3 /^2

(3)

Substituting (3) in (2)
Sx
2

= 2 π

∫a

0

(a^2 /^3 −x^2 /^3 )

[

1 +

y^2 /^3
x^2 /^3

] 1 / 2

dx

= 2 π

∫a

0

(a^2 /^3 −x^2 /^3 )^3 /^2

(

a^2 /^3
x^2 /^3

) 1 / 2

dx

= 2 πa^1 /^3

∫a

0

(a^2 /^3 −x^2 /^3 )^3 /^2 x−^1 /^3 dx

=

6 πa^2
5

∴Sx=

12 πa^2
5

Fig. 1.13Curve of
hypocycloid
x^2 /^3 +y^2 /^3 =a^2 /^3


1.49

∫a

0

∫√a (^2) −x 2
0
(x+y)dydx=
∫a
0


[∫√

a^2 −x^2

0

(x+y)dy

]

dx

=

∫a

0

[(

xy+
y^2
2

)

dx

]√a (^2) −x 2
0


∫a
0


(

x


a^2 −x^2 +

a^2 −x^2
2

)

dx

=

2 a^3
3

1.50 Area to be calculated is


A=ACFD= 2 ×ABED
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