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Math 265: 14.4
Supplemental Instruction
IowaStateUniversity
Trevor
Math 265
Dr. Castillo-Gil
04/15/14
1) Use Green’s Theorem to evaluate the given line integral. Begin by
sketching the region S.
a. ∮ √𝑦 𝑑𝑥 + √𝑥 𝑑𝑦, where C is the closed curve formed by 𝑦 =
0, 𝑥 = 2, and 𝑦 =
𝑥2
2
b. ∮ 𝑥𝑦 𝑑𝑥 + (𝑥 + 𝑦) 𝑑𝑦, where C is the triangle with vertices
(0,0), (2,0), and (0,1)
c. ∮(𝑒 3𝑥 + 2𝑦) 𝑑𝑥 + (𝑥 2 + sin 𝑦) 𝑑𝑦, where C is the rectangle
with vertices (2,1), (6,1), (6,4) and (2,4).
2) Use the formula for A(S) to find the area of the indicated region S.
Make a sketch.
1
a. S is bounded by the curves 𝑦 = 𝑥 3 and 𝑦 = 𝑥 2
2
3) Use the vector forms of Green’s Theorem to calculate a) ∮ 𝐹 ∙ 𝑛 𝑑𝑠 and
b)∮ 𝐹 ∙ 𝑇 𝑑𝑠.
a. 𝐹 = 𝑦 2 𝑖 + 𝑥 2 𝑗; C is the boundary of unit square with vertices
(0,0), (1,0), (0,1), and (1,1)
4) Find the work done by 𝐹 = (𝑥 2 + 𝑦 2 )𝑖 − 2𝑥𝑦𝑗 in moving a body
counterclockwise around the curve C which is the unit square with
vertices (0,0), (1,0), (0,1), and (1,1).
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