374 Higher Engineering Mathematics
6. (a)
2
1
cosec
2 4t dt
(b)
π
2
π
4
(3 sin 2x − 2 cos3x) dx
[(a) 0.2527 (b) 2.638]
7. (a)
1
0
3 e
3t dt (b)
2
−1
2
3 e 2x dx
[(a) 19.09 (b) 2.457]
8. (a)
3
2
2
3x
dx (b)
3
1
2x 2 + 1
x
dx
[(a) 0.2703 (b) 9.099]
9. The entropy change S, for an ideal gas is
given by:
S =
T 2
T 1
C v
dT
T
− R
V 2
V 1
dV
V
where T is the thermodynamic temperature,
V is the volume and R = 8.314. Determine
the entropy change when a gas expands from
1 litre to 3 litres for a temperature rise from
100 K to 400 K given that:
C v = 45 + 6 × 10
−3 T + 8 × 10
−6 T
2
.
[55.65]
10. The p.d. between boundaries a and b of an
electric field is given by: V =
b
a
Q
2πrε 0 ε r
dr
If a = 10, b = 20, Q =2 × 10 −6 coulombs,
ε 0 = 8.85 ×10 −12 and ε r = 2.77, show that
V = 9 kV.
11. The average value of a complex voltage waveform is given by:
V AV =
1
π
π
0
(10 sin ωt + 3 sin3ωt
+ 2 sin 5ωt) d(ωt)
Evaluate V AV correct to 2 decimal places.
[7.26]
6. (a)
2
1
cosec
2 4t dt
(b)
π
2
π
4
(3 sin 2x − 2 cos3x) dx
[(a) 0.2527 (b) 2.638]
7. (a)
1
0
3 e
3t dt (b)
2
−1
2
3 e 2x dx
[(a) 19.09 (b) 2.457]
8. (a)
3
2
2
3x
dx (b)
3
1
2x 2 + 1
x
dx
[(a) 0.2703 (b) 9.099]
9. The entropy change S, for an ideal gas is
given by:
S =
T 2
T 1
C v
dT
T
− R
V 2
V 1
dV
V
where T is the thermodynamic temperature,
V is the volume and R = 8.314. Determine
the entropy change when a gas expands from
1 litre to 3 litres for a temperature rise from
100 K to 400 K given that:
C v = 45 + 6 × 10
−3 T + 8 × 10
−6 T
2
.
[55.65]
10. The p.d. between boundaries a and b of an
electric field is given by: V =
b
a
Q
2πrε 0 ε r
dr
If a = 10, b = 20, Q =2 × 10 −6 coulombs,
ε 0 = 8.85 ×10 −12 and ε r = 2.77, show that
V = 9 kV.
11. The average value of a complex voltage waveform is given by:
V AV =
1
π
π
0
(10 sin ωt + 3 sin3ωt
+ 2 sin 5ωt) d(ωt)
Evaluate V AV correct to 2 decimal places.
[7.26]