56
1 Mathematical Physics
Fig. 1.15a Area bounded by ABGFA (see the text, Prob 1.53a )
Fig. 1.15b Volume of the
cylinder plus the cone
(See Prob 1.15b)
= π. 2
2
5 +
4
3
= 25
1
3
π units
1.54 (a) Area =
2π
0
ydx =
π
0
x sin x dx +
2π
π
x sin x dx
= −x cos x + sin x|
π
0 − x cos x + sin x|
2π
π = π + 3π = 4π
The area refers to the magnitude
(b) Volume, V = 2π
y
2 dx = 2π
x
2 sin
2 x dx
=
4π
4
3
−
π
2
2
1 Mathematical Physics
Fig. 1.15a Area bounded by ABGFA (see the text, Prob 1.53a )
Fig. 1.15b Volume of the
cylinder plus the cone
(See Prob 1.15b)
= π. 2
2
5 +
4
3
= 25
1
3
π units
1.54 (a) Area =
2π
0
ydx =
π
0
x sin x dx +
2π
π
x sin x dx
= −x cos x + sin x|
π
0 − x cos x + sin x|
2π
π = π + 3π = 4π
The area refers to the magnitude
(b) Volume, V = 2π
y
2 dx = 2π
x
2 sin
2 x dx
=
4π
4
3
−
π
2
2