1.) <span>the ratio of the note A to middle C = 440.0 / 261.6 = 1.6820 ≈ 1.6818
2.) </span><span>the ratio of the note D to middle C = 293.7 / 261.6 = 1.1227
3.) </span><span>D# = 293.6 x 1.0595 = 311.1
4.) </span><span>the ratio of the frequency of G to C is 1 : 262/392 = 1 : 0.6683 ≈ 3 : 2
5.) </span><span>the ratio of the frequency of E to C = 1 : 262/330 = 1 : 0.7939 ≈ 5 : 4
6.) The </span><span>component in a musical note is referred to as the pitch of a sound is the frequency.</span>
he volume of the solid under a surface
z
=
f
(
x
,
y
)
and above a region D is given by the formula
∫
∫
D
f
(
x
,
y
)
d
A
.
Here
f
(
x
,
y
)
=
6
x
y
. The inequalities that define the region D can be found by making a sketch of the triangle that lies in the
x
y
−
plane. The bounding equations of the triangle are found using the point-slope formula as
x
=
1
,
y
=
1
and
y
=
−
x
3
+
7
3
.
Here is a sketch of the triangle:
Intersecting Region
The inequalities that describe D are given by the sketch as:
1
≤
x
≤
4
and
1
≤
y
≤
−
x
3
+
7
3
.
Therefore, volume is
V
=
∫
4
1
∫
−
x
3
+
7
3
1
6
x
y
d
y
d
x
=
∫
4
1
6
x
[
y
2
2
]
−
x
3
+
7
3
1
d
x
=
3
∫
4
1
x
[
y
2
]
−
x
3
+
7
3
1
d
x
=
3
∫
4
1
x
[
49
9
−
14
x
9
+
x
2
9
−
1
]
d
x
=
3
∫
4
1
40
x
9
−
14
x
2
9
+
x
3
9
d
x
=
3
[
40
x
2
18
−
14
x
3
27
+
x
4
36
]
4
1
=
3
[
(
640
18
−
896
27
+
256
36
)
−
(
40
18
−
14
27
+
1
36
)
]
=
23.25
.
Volume is
23.25
.
