5.9 Exercises 161
Find the average values in the given intervals:
- 2 x
2
1 + 13 x 1 + 1 4; − 11 ≤ 1 x 1 ≤ 1 + 1 27.cos 13 θ;0 1 ≤ 1 θ 1 ≤ 1 π 22 28.1; 3 1 ≤ 1 x 1 ≤ 15
Demonstrate and sketch a graph to interpret:
- (i) Show that .(ii) Calculate , ,
. (iii)Sketch a graph to interpret these results.
Evaluate and sketch a graph to interpret:
3 5. (i)Show that ,(ii) evaluate.
Evaluate:
For each function, state if it is an even function of x, an odd function, or neither. If neither,
give the even and odd components.
40.sin 12 x 41.cos 13 x 42.sin 1 x 1 cos 1 x 43.x 44.x
4
- 3 x
2
1 + 12 x 1 + 11
46.e
−x47.(3x
2
1 + 12 x 1 + 1 1)e
−xSection 5.4
48.The equation of an ellipse with centre at the origin is , where, ifa 1 > 1 b,ais
the major axis and bthe minor axis (ifa 1 = 1 b, we have a circle). Use Method 1 in
Example 5.11 to find the area of the ellipse.
49.Find the length of the curve betweenx 1 = 10 andx 1 = 11.
Section 5.6
50.Three masses,m
1
1 = 11 ,m
2
1 = 12 andm
3
1 = 13 , lie on a straight line withm
1
atx
1
1 = 1 − 4 ,m
2
at
x
2
1 = 1 − 1 andm
3
atx
3
1 = 1 + 4 with respect to a point Oon the line. Calculate (i)the position
of the centre of mass, (ii)the moment of inertia with respect to O, and (iii)the moment
of inertia with respect to the centre of mass.
51.The distribution of mass in a straight rod of length lis given by the density function
ρ(x) 1 = 1 x
2
; 01 ≤ 1 x 1 ≤ 1 l. Find (i)the total mass, (ii)the mean density, (iii)the centre of mass,
yx=
1
2
32
x
a
y
b
2
2
2
2
+= 1
Z
2
2
1
∞
dx
xx()
−Z
2
1
∞
dx
xx()−
Z
0
2
∞
edx
−xZ
0
3
∞
edt
−tZ
0
1
lnxdx
d
dx
(ln ) lnxxx−=x
Z
−
+
−
+
=
−≤
aa
xxfxdx fx
ex
ex
() ()where
if
if
0
0
Z
−
+
=
−≤
11
0
0
fxdx fx
xx
xx
() ()where
if
if
Z
−
+
=
+<
≥
1
3
2
2
21
1
fxdx fx
xx
xx
() ()where
if
if
Z
−ππcosxdx
Z
−ππ 2Z cosxdx
−π0
ZZ cosxdx
0
2
2
πππcosxdx=− cosxdx
ZZZ
2
3
2
6
3
6
edx edx edx
xxxZZZZ =−
0
3
0
1
1
2
2
3
edx edx edx edx
−−−−xxxx=++