CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND 3  d  tan 2sin cos d  2 a  0 0 2   0  a   0 3  0 0  3  a  0  d a 3 VOLUME INTEGRALS V S If is a closed region bounded by a surface and is a vector field at each point F V S, of and on its boundary surface then  FdV V is the volume integral of throughout the region. In most standard texts, triple F integrals are used as notation for volume integral. Z Example 5.15 Evaluate where V is the region bounded by the FdV dV x=0, x=2, y=0, y=3, z=0, z=4, planes and F  xyiˆ  zˆj x2kˆ 4   F xyiˆ zˆj x2kˆ;  dV dxdydz Y 2 3 4 3 2 V O 0 0 0 X 4 3 4 0 0 0 Example 5.16 V z=0, Evaluate where and is the region bounded by the planes  2iˆ  2zˆj  FdV F ykˆ z=4 and the surface . 9   x2 y 2
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS  a 3   0 0 2  tan 2sin cos d d  a  a 3 3   0 0   0 2 d  a 3   0  0 VOLUME INTEGRALS If V is a closed region bounded by a surface S and F is a vector field at each point of V and on its boundary surface S, then  FdV V is the volume integral of F throughout the region. In most standard texts, triple integrals are used as notation for volume integral. Z Example 5.15 Evaluate FdV where V is the region bounded by the planes x=0, x=2, y=0, y=3, z=0, z=4, and F  xyiˆ  zˆj x2kˆ 4 dV F  xyiˆ  zˆj x2kˆ; dV  dxdydz V 0 0 0 4 3 2 2 O 3 Y X 4 3 4 0 0 0 Example 5.16 Evaluate FdV where F  2iˆ  2zˆj  ykˆ and V is the region bounded by the planes z=0, z=4 and the surface x2  y 2  9 .