If a , b , and c are positive constants, then the transformation x = a u , y = b υ , z = c w can be rewritten as x / a = u , y / b = υ , z / c = w , and hence it maps the spherical region u 2 + υ 2 + w 2 ≤ 1 into the ellipsoidal region x 2 a 2 + y 2 b 2 + z 2 c 2 ≤ 1 In these exercises, perform the integration by transforming the ellipsoidal region of integration into a spherical region of integration and then evaluating the transformed integral in spherical coordinates. ∭ G x 2 d V , where G is the region enclosed by the ellipsoid 9 x 2 + 4 y 2 + z 2 = 36.
If a , b , and c are positive constants, then the transformation x = a u , y = b υ , z = c w can be rewritten as x / a = u , y / b = υ , z / c = w , and hence it maps the spherical region u 2 + υ 2 + w 2 ≤ 1 into the ellipsoidal region x 2 a 2 + y 2 b 2 + z 2 c 2 ≤ 1 In these exercises, perform the integration by transforming the ellipsoidal region of integration into a spherical region of integration and then evaluating the transformed integral in spherical coordinates. ∭ G x 2 d V , where G is the region enclosed by the ellipsoid 9 x 2 + 4 y 2 + z 2 = 36.
If a, b, and c are positive constants, then the transformation
x
=
a
u
,
y
=
b
υ
,
z
=
c
w
can be rewritten as
x
/
a
=
u
,
y
/
b
=
υ
,
z
/
c
=
w
,
and hence it maps the spherical region
u
2
+
υ
2
+
w
2
≤
1
into the ellipsoidal region
x
2
a
2
+
y
2
b
2
+
z
2
c
2
≤
1
In these exercises, perform the integration by transforming the ellipsoidal region of integration into a spherical region of integration and then evaluating the transformed integral in spherical coordinates.
∭
G
x
2
d
V
,
where G is the region enclosed by the ellipsoid
9
x
2
+
4
y
2
+
z
2
=
36.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Give two polar representations for the point (x, y) = (0, 1), one with negative r and one with positive r.
II. Evaluate in rectangular form (a + jb) and in polar form (rZ0) given:
Z1 = -5 + j12,Z2 = 3 + j4,Z3 = -2 – j,Z4 = 1 – j3.
1. Z,Z,Z3Z4
Z,Z3
2.
Z2+Z4
Z,Z2_ ( Z3Z4
3.
Z3+Z4 \Z1-Z2,
219. The diagram at right shows a rectangular solid, two of whose
vertices are A = (0, 0, 0) and G = (4,6,3).
→
(a) Find vector projections of AG onto AB, AD, and AE.
(b) Find the point on segment AC that is closest to the midpoint
of segment GH.
220. The result of reflecting across the line y
= -x and then
rotating 330 degrees counterclockwise around the origin is an
F
B
Z
E
A
G
C
H
D
-Y
Precalculus: Mathematics for Calculus - 6th Edition
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