MATH 2423 #15: QUESTION 2

#15: 2a) ANSWER: - 4pi
SOLUTION: The boundary is the intersection of x ² + y ² + z ² = 4 and z = 0 which is x ² + y ² = 4 in the xy-plane.

• The equation of the boundary is r (t) = < 2 cost , 2 sint , 0 >. As t goes from 0 to 2pi, the points on the curve go counterclockwise.
• dr = 2 < - sint , cost , 0 > dt
• On the curve, x = 2 cost, y = 2sint and z = 0.
• F = < y , x ² + z , xyz > = < 2sint , 4cos ² t , 0 > = 2 < sint , 2cos ² t , 0 >
• F • dr = 4 ( - sin ² t + 2cos ³ t ) dt
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© 2000: Roxanne M. Byrne