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.
Precalculus: Mathematics for Calculus - 6th Edition
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