# HW 12 - Math E-21a Fall 2009 HW #12 problems Problems due...

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1 Math E-21a – Fall 2009 – HW #12 problems Problems due Thurs, Dec 3 : Section 13.4 : 4. Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. C x dx y dy , C consists of the line segments from (0,1) to (0,0) and from (0,0) to (1,0) and the parabola 2 1 y x  from (1,0) to (0,1). 8. Use Green’s Theorem to evaluate the line integral 22 3 4 C xyd x x yd y along the positively oriented curve C that is the triangle with vertices (0,0), (1,3), and (0,3). 12. Use Green’s Theorem to evaluate the line integral sin cos C ydx x ydy along the positively oriented curve C that is the ellipse 1 xx y y  . 14. Use Green’s Theorem to evaluate C Fd r where (, ) c o s, 2 s i n x yy x xy x  F and C is the triangle from (0,0) to (2,6) to (2,0) to . 18. A particle starts at the point (2 ,0 ) , moves along the x -axis to , and then along the semicircle 2 4 y x to the starting point. Use Green’s Theorem to find the work done on this particle by the force field 32 , 3 x yx y F . 22. Let D be a region bounded by a simple closed path C in the xy -plane. Use Green’s Theorem to prove that the coordinates of the centroid x y of D are 11 CC x xdy y ydx AA   where A is the area of D .

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## This note was uploaded on 02/10/2011 for the course MATH E-21a taught by Professor - during the Fall '09 term at Harvard.

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HW 12 - Math E-21a Fall 2009 HW #12 problems Problems due...

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