265Lesson23Problems

# 265Lesson23Problems - x 2 y 2 = 1 and x 2 y 2 = 4 Let F x,y...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 23 PROBLEMS Problems are worth 20 points each. (1) Use Green’s Theorem to evaluate the line integral I C xy d x + y d y where C is the unit circle orientated counterclockwise. (2) Use Green’s Theorem to evaluate the line integral I C (ln x + y ) d x - x 2 d y over the rectangle in the xy -plane with vertices at (1 , 1) , (3 , 1) , (1 , 4), and (3 , 4). (3) If C is a simple closed curve, what is the value of I C y d x + x d y ? (4) Use Green’s Theorem to evaluate the line integral of the vector ﬁeld -→ F ( x,y ) = x 3 -→ ı + x -→ around the unit square (the square in the xy -plane with vertices at (0 , 0) , (0 , 1) , (1 , 0), and (1 , 1)) orientated clockwise. (5) Let A be the region in the xy -plane between the circles
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Unformatted text preview: x 2 + y 2 = 1 and x 2 + y 2 = 4. Let- F ( x,y ) = h-y 3 , 2 i . Use Greens Theorem to evaluate I C- F d- s where C is the boundary of A with the outer circle orientated counterclockwise and the inner circle orientate clockwise (in other words, with the entire boundary of A orientated in the positive direction). (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Use Greens Theorem to compute the area above the x-axis and under one arch of the cycloid given parametrically by x = f ( t ) = t-sin t, y = g ( t ) = 1-cos t , t 2 . 1...
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