MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Problem Set 9 Due Thursday 11 8 07 12 45 pm 10 points Part A Hand in the underlined problems only the others are for more practice Lecture 24 Thu Nov 1 Green s theorem Read 15 4 to top of p 1043 Work 4D 1abc 2 3 4 5 Lecture 25 Fri Nov 2 Flux Normal form of Green s theorem Read Notes V3 V4 Work 4E 1ac 2 3 4 5 don t use Green s theorem 4F 3 4 Lecture 26 Tue Nov 6 Read Notes V5 Lecture 27 Thu Nov 8 Simply connected regions Review Exam 3 covering lectures 18 26 16 points Part B Directions Attempt to solve each part of each problem yourself If you collaborate solutions must be written up independently It is illegal to consult materials from previous semesters With each problem is the day it can be done Write the names of all the people you consulted or with whom you collaborated and the resources you used Problem 1 Thursday 4 points 2 2 a Show that if a simple closed positively oriented curve C is the boundary of a region R then Area R C x dy C y dx b Use this result to calculate the area of the region between the x axis and one arch of the cycloid with parametric equations x a t sin t y a 1 cos t Hint the second line integral leads to easier calculations Problem 2 Thursday 6 points 4 2 a For what simple closed positively oriented curve C in the plane does the line integral x2 y y 3 y dx 3x 2y 2 x ey dy have the largest positive value use Green s C theorem b Determine what this value is Problem 3 Friday 6 points 2 2 2 a Let C be the unit circle oriented counterclockwise and consider the vector eld F xy y 2 Which portions of C contribute positively to the ux of F Which portions contribute negatively b Find the ux of F through C by direct calculation evaluating a line integral Explain your answer using a c Find the ux of F through C using Green s theorem 1
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