1 Evaluate Solutions to Midterm 2 I 1 2 ydx xdy x2 y2 C a counterclockwise around the circle x2 y2 a2 b counterclockwise around the boundary of the region 1 x2 y2 2 y 0 Hint Does the Green s Theorem work for both a and b Z 2 0 1 2 dt 1 Solution a Let x a cos t y a sin t for t 0 2 Then I 1 2 C ydx xdy x2 y2 1 2 Z 2 0 b Let D be the domain enclosed by C Let P a sin t a sin t dt a2 cos2 tdt a2 y x2 y2 Q Q x y2 x2 x2 y2 2 P y y2 x2 x2 y2 2 I 1 2 C ydx xdy x2 y2 1 2 P y dA 0 ZZ cid 18 Q x D x x2 y2 Q x cid 19 By Green Theorem 1 2 a Find a function f such that b Evaluate the line integralR f F x y z y2 4xz i 2xy 2yz j y2 2x2 k C F dr where C is the line segment from 1 0 2 to 4 0 3 y2 4xz implies f x y z R y2 4xz dx g y z xy2 2zx2 g x y Solution a f x f y f y So 2xy g y But 2xy 2yz so g y 2yz g y z R 2yzdy h z zy2 h z Thus f x y z xy2 2x2z h z and y2 2x2 h0 z f z y2 2x2 so h0 z 0 h z K Hence f x y z xy2 2zx2 zy2 taking K 0 f z But b R C F dr f 4 0 3 f 1 0 2 16 6 4 100 2 3 Label each expression as a scalar quantity a vector quantity or unde ned if f is a scalar function and F is a vector eld 1 f 2 f 3 f 4 F 5 F 6 F 7 F 8 F 9 F 10 F Solution 1 unde ned 2 scalar quantity 3 vector quantity 4 unde ned 5 vector quantity 6 scalar quantity 7 unde ned 8 unde ned 9 unde ned 10 vector quantity 3 4 a If C is the line segment connecting the point x1 y1 to the point x2 y2 show that Z C xdy ydx x1y2 x2y1 b If the vertices of a polygon in counterclockwise order are x1 y1 x2 y2 xn yn show that the area of the polygon is A x1y2 x2y1 x2y3 x3y2 xn 1yn xnyn 1 xny1 x1yn 1 2 c Find the area of the pentagon with vertices 0 0 2 1 1 3 0 2 and 1 1 Solution a We write parametric equations of the line segment as x 1 t x1 tx2 y 1 t y1 ty2 0 t 1 Then dx x2 x1 dt and dy y2 y1 dt so xdy ydx 1 t x1 tx2 y2 y1 dt 1 t y1 ty2 x2 x1 dt x1 y2 y1 y1 x2 x1 t y2 y1 x2 x1 x2 x1 y2 y1 dt x1y2 x2y1 dt x1y2 x2y1 0 0 Z 1 Z 1 Z 1 C xdy ydx RR ZZ cid 18 Z 0 dA 1 2 xdy ydx D 1 2 C1 Z Z C C2 b We apply Green s Theorem to the path C C1 C2 Cn where Ci is the line segment that joints xi yi to xi 1 yi 1 for i 1 2 n 1 and Cn is the line segment that joints xn yn to x1 y1 Thus 1 2 D dA where D is the polygon bounded by C Therefore R area of polygon A D xdy ydx xdy ydx xdy ydx xdy ydx Z Cn 1 Z Cn cid 19 To evaluate these integrals we use the formula from a to get A D x1y2 x2y1 x2y3 x3y2 xn 1yn xnyn 1 xny1 x1yn Z C c A 0 1 2 0 2 3 1 1 1 2 0 3 0 1 1 2 1 0 0 1 1 2 1 2 1 2 0 5 2 2 9 2 4 5 Find a parametric representation for the surface a The plane that passes through the point 5 1 0 and contains the vectors h4 1 2i and h5 2 3i b The part of the cylinder y2 z2 16 that lies between the planes x 0 and x 5 Solution a Parametric equations for the plane through the point 5 1 0 that contains the vectors a h4 1 2i and b h5 2 3i are x 5 u 4 v 5 5 4u 5v y 1 u 1 v 2 1 u 2v z 0 u 2 v 3 2u 3v b Parametric equations are x x y 4 cos z 4 sin 0 x 5 0 2 5
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