Exercises

Use (1) or (2) to evaluate the following double integrals

1. ó õ 1 -1  ó õ Ö 1-x2 dydx -Ö 1-x2 2. ó õ 1 -1  ó õ Ö 1-x2 xdydx -Ö 1-x2 3. ó õ 2 -2  ó õ Ö 4-x2  ydydx -Ö 4-x2 4. ó õ 2 -2  ó õ Ö 4-x2 ( x+y) dydx -Ö 4-x2 5. ó õ 1 -1  ó õ Ö 1-x2  y 2-x dydx -Ö 1-x2 6. ó õ 1 -1  ó õ Ö 1-x2  x 2-y dydx -Ö 1-x2

Use Green's theorem to evaluate the following line integrals.

7.  1 2 ¶R  xdy-ydx, 8. ¶R  y2dx-x2dy, R is the unit square R is the unit square 9. ¶D  x2dy 10. ¶D  y2dx 11. ¶R  sin( x3) dx 12. ¶R  exdy+eydx R is the unit square R is the unit square 13. ¶R  2xydx+x2dy 14. ¶R  exdx+eydy R is the unit square R is the unit square 15. ¶D  x2dy-2xydx 16. ¶D  (x2+y2) dx+2xydy 17. ¶R  sin( py) dx 18. ¶D  dx+dy x+y R between y = 0 and y = x for x in [0,1] 19. ¶R  x2dy-2ydx 20. ¶R  (x2+y2) ( dx+dy) R between y = 0 and y = 1-x2 R between y = 0 and y = 1-x2

Use theorem 4 to find the area of the regions bound by the following curves.

21. C: r( t) = át3-t,t2-t ñ 22. C: r( t) = á t3-t2,t2-t ñ t in [ 0,1] t in [ 0,1] 23. C: r( t) = á sin( pt),t3-t ñ 24. C: r( t) = á t3-t,t2 ñ t in [ 0,1] t in [ 0,1] 25. C: r( t) = á sin( 2t),sin( 3t) ñ 26. C: r(t) = á 3cos( 2t) ,4cos( 2t) ñ t in [ 0,p] t in [ 0,p] 27. C: r( t) = á sin3( t),cos3( t) ñ 28. C: r(t) = á cos( t) ,tsin( t) ñ t in [ 0,2p] t in [ 0,2p]

In exercises 29-30, determine the orientation of the boundary and then use Green's theorem to find the area of the given region.

29. 30. C1: r( t) =   2( 4-t2)5/4-2,    -5 2 t ( 4-t2) 1/4 , t in [ -2,2] C1: r( t) = á 4cos( t) ,4sin( t) ñ , t in [ 0,2p] C2: r( t) = á 6( 1-t2)3/2,  -3t( 1-t2) 1/2 ñ , t in [-1,1] C2: r( t) = á2sin2( t) +1,2sin( 2t) ñ , t in [ 0,p] C3: r( t) = á 2sin2( t) -3,2sin( 2t) ñ , t in [ 0,p]

 31. In polar coordinates, the function 

r =  1 2-cos( q)

is an ellipse with one focus at the origin.  Explain why this ellipse has a parametrization of

r( q) =  cos( q) 2-cos( q) ,  sin( q) 2-cos( q)

for q in [ 0,2p] and then find its area.

32. Repeat exercise 31 with the general form of an ellipse in polar coordinates

r =  p 1-ecos( q)

where p > 0 and | e| < 1.

33. Set up the area integral for the region R inside the ellipse

r( q) =  cos( q) 2-cos( q) ,  sin( q) 2-cos( q)

for q in [ 0,2p] and outside the circle centered at the origin with radius 0.1. What is the area of R?

34. Use (1) or (2) to evaluate

óó õõ   R  x2dA

where R is the annulus centered at the origin with inner radius 1 and outer radius 3.

35. Finish the special case of Green's theorem by proving (2). That is, show that if R is a type II region, then

óó õõ   R NxdA = ¶R  N( x,y) dx

36. Show that if f( x,y) satisfies Laplace's equation,

¶2f ¶x2 +  ¶2f ¶y2 = 0

on a simply-connected region R, then for all closed curves C in R we have

C  ( fydx-fxdy) = 0

(Conversely, it can be shown that if the line integral above is 0 for any closed curve C, then f satisfies Laplace's equation).

37. Show that the coordinates of the centroid of a simply connected region R are given by

x  =    1 2A ¶R  x2dy,        y  =    -1 2A ¶R  y2dx

where A is the area of R.

38. Use the result in exercise 37 to find the centroid of the region in example 4, in which the region is inside the curve r( t) = á t⁴-t²,t⁶-t² ñ , t in [0,1] . (You might also want to find the centroids of some of the region enclosed by the curves in exercises 121-30).

39. Write to Learn: Write a short essay which shows that if R is a simply connected region and if F( x,y) = áM( x,y) ,N( x,y) ñ , then

¶R  F·dr =    óó õõ   R  curl( F) ·k  dA

In addition, interpret the result and what it says about the curl of a 2-dimensional vector field.

40. Write to Learn: Let R be a triangle with vertices (x₁,y₁) , ( x₂,y₂) , and (x₃,y₃) . Apply each of the results in theorem 2 to find three different formulas for finding the area of a triangle given the coordinates of its vertices. Explain your results in a short essay and apply each result to the triangle with vertices ( 1,2) , ( 7,1) , and ( 5,4) , respectively.