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Line integrals and rectangles

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Line Integrals

Please see the attached. Please do the problem(s) in detail and show all work.
This question requires a line integral around the rectangle defined by the points (1,-1), (1,1), -1,1), (-1,-1) and with the function given. This defines 4 integrals that have to be evaluated as described in the problem.

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Solution Summary

An integral along a rectangle of a function is solved in the solution.

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I = Integral(C) [((x-a).dy - y.dx)/((x-a)^2 +y^2)] = I1 + I2 + I3 + I4

where,

I1 = Integral(y=-1 to y =1) [(1-a) . dy / ((1-a)^2 +y^2)] (Because, x = 1)

I2 = Integral(x=1 to x=-1) [ - dx / ((x-a)^2 +1)] (Because, y = 1)

I3 = Integral(y = 1 to y = -1) [ (-1-a) . dy / ((-1-a)^2 +y^2)] (Because, x = -1)

I4 = Integral(x=-1 to x=1) [ dx / ((x-a)^2 + 1)] (Because, y = -1)

Solution:
I1 = (1-a) . (1/(1-a)) . tan-1( y / (1-a) ) | (y=-1 to y = 1)

= tan-1( 1 / (1-a) ) - tan-1( -1 / (1-a) ...

Solution provided by:
Amrit Lal Ahuja, PhD (IP)
Education
  • BEng, Allahabad University, India
  • MSc , Pune University, India
  • PhD (IP), Pune University, India
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