Calculus- Early Transcendentals, 2021a

16.7. Surface Integrals 563
=
=
∫ 2π ∫ 2
0 0
∫ 2π ∫ 2
0 0
v 2 cos 2 u + v 2 sin 2 u − v 3 dvdu
v 2 − v 3 dvdu = − 8π 3 .
♣
Exercises for 16.7
Exercise 16.7.1 Find the center of mass of an object that occupies the upper hemisphere of x 2 +y 2 +z 2 = 1
and has density x 2 + y 2 .
Exercise 16.7.2 Find the center of mass of an object that occupies the surface z = xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
and has density √ 1 + x 2 + y 2 .
Exercise 16.7.3 Find the center of mass of an object that occupies the surface z = √ x 2 + y 2 , 1 ≤ z ≤ 4
and has density x 2 z.
Exercise 16.7.4 Find the centroid of the surface of a right circular cone of height h and base radius r, not
including the base.
∫∫
Exercise 16.7.5 Evaluate 〈2,−3,4〉·NdS, where D is given by z = x 2 + y 2 , −1 ≤ x ≤ 1, −1 ≤ y ≤ 1,
D
oriented up.
∫∫
Exercise 16.7.6 Evaluate 〈x,y,3〉·NdS, where D is given by z = 3x − 5y, 1 ≤ x ≤ 2, 0 ≤ y ≤ 2,
D
oriented up.
∫∫
Exercise 16.7.7 Evaluate 〈x,y,−2〉·NdS, where D is given by z = 1 − x 2 − y 2 ,x 2 + y 2 ≤ 1, oriented
D
up.
∫∫
Exercise 16.7.8 Evaluate 〈e x ,e y ,z〉·NdS, where D is given by z = xy, 0 ≤ x ≤ 1, −x ≤ y ≤ x, oriented
D
up.
∫∫
Exercise 16.7.9 Evaluate 〈xz,yz,z〉·NdS, where D is given by z = a 2 −x 2 −y 2 ,x 2 +y 2 ≤ b 2 ,oriented
D
up.
Exercise 16.7.10 A fluid has density 870 kg/m 3 and flows with velocity v = 〈z,y 2 ,x 2 〉, where distances are
in meters and the components of v are in meters per second. Find the rate of flow outward through the
portion of the cylinder x 2 + y 2 = 4, 0 ≤ z ≤ 1 for which y > 0.
Exercise 16.7.11 Gauss’s Law says that the net charge, Q, enclosed by a closed surface, S, is
Q = ε 0
∫∫
e · NdS