CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Z Since we have a cylinder, we convert coordinate system 4 into the cylindrical one. dV Thus we have   where  cos,  sin,  x y z z 0  2, 0  3, 0 4 z 3 Y O  3 Hence X 2ˆ dxdydz 2zˆ    yk V 2 3 4 2 3 4 0 0 0 0 0 z0 3 0 2 3 2 0 0  3 2 iˆ 2 2 sin  72iˆ 2   kˆd   ˆj ˆj 0 Example 5.17 2x Evaluate where is the closed region bounded by the cylinder and the  ydV V planes x=0, y=0, y=z and z=0 We have 2 2 Example 5.18 Evaluate where V is the region bounded by z=x2+y2 and  x2 y dxdydz, 2 V
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Z Since we have a cylinder, we convert coordinate system into the cylindrical one. 4 dV Thus we have x   cos, y   sin, z  z where 0  2, 0  3, 0 z 4 O  3 Y 3 Hence  2ˆ  2zˆ  yk dxdydz V X 2 3 4 0 0 z0 2 3 4 0 0 0 2 0 3 0 2 2 0  3 3 2 0 iˆ  2 ˆj  sin  kˆd  72iˆ  2 ˆj Example 5.17 Evaluate 2x  ydV where V is the closed region bounded by the cylinder and the planes x=0, y=0, y=z and z=0 We have 2 2 Example 5.18 Evaluate V x2  y 2 dxdydz, where V is the region bounded by z=x2+y2 and