12. The ellipses 3x 2 + 4y 2 = C all satisfy the differential equation
6x + 8y
dy
= 0,
dx
or
dy
3x
=− .
dx
4y
A family of curves that intersect these ellipses at right angles must therefore have slopes
dy
4y
given by
=
. Thus
dx
3x
Z
Z
dx
dy
=4
3
y
x
3 ln |y| = 4 ln |x| + ln |C|.
The family is given by y 3 = C x 4 .
8.
y
1
3
x
Fig. R-8
Let the disk have centre (and therefore centroid) at (0, 0). Its area is 9π . Let the hole have
centre (and therefore centroid) at (1, 0). Its area is π . The remaining part has area 8π and
centroid at (x, 0), where
(9π )(0) = (8π )x + (π )(1).
Thus x = −1/8. The centroid of the remaining part is 1/8 ft from the centre of the disk on
the side opposite the hole.
20.
y ′ + (cos x)y = 2xe− sin x ,
Z
µ = cos x d x = sin x
y(π ) = 0
d sin x
(e
y) = esin x (y ′ + (cos x)y) = 2x
dx
Z
esin x y =
2x d x = x 2 + C
y(π ) = 0 ⇒ 0 = π 2 + C ⇒ C = −π 2
y = (x 2 − π 2 )e− sin x .
17.
1
2
2
√ e−(x−µ) /2σ
σ 2π Z
∞
1
2
2
xe−(x−µ) /2σ d x
mean = √
σ 2π −∞
f µ,σ (x) =
Let z =
dz =
=
=
variance =
=
=
x −µ
σ
1
dx
σ
Z ∞
1
2
(µ + σ z)e−z /2 dz
√
2π Z−∞
∞
µ
2
e−z /2 dz = µ
√
2π −∞
E (x − µ)2
Z ∞
1
2
2
(x − µ)2 e−(x−µ) /2σ d x
√
σ 2π Z−∞
∞
1
2
σ 2 z 2 e−z /2 dz = σ Var(Z ) = σ
√
σ 2π −∞
9. A layer of water between depths y and y + dy has volume d V = π(a 2 − y 2 ) dy and weight
d F = 9, 800π(a 2 − y 2 ) dy N. The work done to raise this water to height h m above the
top of the bowl is
dW = (h + y) d F = 9, 800π(h + y)(a 2 − y 2 ) dy N·m.
Thus the total work done to pump all the water in the bowl to that height is
Z
a
(ha 2 + a 2 y − hy 2 − y 3 ) dy
hy 3
y 4 a
a2 y 2
2
−
−
= 9, 800π ha y +
2
3
4 0
3
a4
2a h
+
= 9, 800π
3
4
8h
3
3 3a + 8h
= 2450π a a +
N·m.
= 9, 800π a
12
3
W = 9, 800π
0
y
dy
Fig. 6-9
a