Higher Engineering Mathematics, Sixth Edition

374 Higher Engineering Mathematics

6. (a)

∫ 2

1

cosec^24 tdt

(b)

∫ π

2

π

4

(3sin2x−2cos3x)dx

[(a) 0.2527 (b) 2.638]

7. (a)

∫ 1

0

3e^3 tdt (b)

∫ 2

− 1

2

3e^2 x

dx

[(a) 19.09 (b) 2.457]

8. (a)

∫ 3

2

2

3 x

dx (b)

∫ 3

1

2 x^2 + 1

x

dx

[(a) 0.2703 (b) 9.099]

9. The entropy changeS, for an ideal gas is
   given by:

S=

∫T 2

T 1

Cv

dT

T

−R

∫V 2

V 1

dV

V

whereTis the thermodynamic temperature,

V is the volume andR= 8 .314. Determine

the entropy change when a gas expands from

1litre to 3litres for a temperature rise from

100K to 400K given that:

Cv= 45 + 6 × 10 −^3 T+ 8 × 10 −^6 T^2.

[55.65]

10. The p.d. between boundariesaandbof an

electric field is given by:V=

∫b

a

Q

2 πrε 0 εr

dr

If a=10, b=20, Q= 2 × 10 −^6 coulombs,

ε 0 = 8. 85 × 10 −^12 and εr= 2 .77, show that

V=9kV.

11. The average value of a complex voltage wave-
    form is given by:

VAV=

1

π

∫π

0

(10sinωt+3sin3ωt

+2sin5ωt)d(ωt)

EvaluateVAVcorrect to 2 decimal places.

[7.26]