![]() ![]() ∫ C v ⋅ N ds = ∬ D ( P x + Q y ) d A = ∬ D 8 d A = 8 ( area of D ) = 80. Therefore, by the same logic as in Example 6.40, ![]() Let D be any region with a boundary that is a simple closed curve C oriented counterclockwise. The logic of the previous example can be extended to derive a formula for the area of any region D. In Example 6.40, we used vector field F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 to find the area of any ellipse. Therefore, the area of the ellipse is π a b. d r = 1 2 ∫ C − y d x + x d y = 1 2 ∫ 0 2 π − b sin t ( − a sin t ) + a ( cos t ) b cos t d t = 1 2 ∫ 0 2 π a b cos 2 t + a b sin 2 t d t = 1 2 ∫ 0 2 π a b d t = π a b. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |