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Title: Evaluate the given integral by making an appropriate change of variables, where R is the trapezoidal region? Post by: rkahn on Jul 31, 2013 Evaluate the given integral by making an appropriate change of variables, where R is the trapezoidal region with vertices (2, 0), (10, 0), (0, 10), and (0, 2).
L=double integral of 5(cos(3(y-x)/(y+x))) dA Title: Re: Evaluate the given integral by making an appropriate change of variables, where R is the trapezo Post by: doseofmegan on Aug 2, 2013 "First of all, note that the edges of the trapezoid have equations
x = 0, y = 0, x+y = 2, and x+y = 10. This and the integrand suggests the substitution u = y - x, and v = y + x. So, x = (-u+v)/2 and y = (u+v)/2. The boundaries of R transform as follows: x = 0 ==> u = v, and y = 0 ==> u = -v, while x+y = 2 and x+y = 10 ==> v = 2, v = 10. Moreover, ∂(x,y)/∂(u,v) = |-1/2 1/2| |1/2 1/2| = -1/2. So, ∫∫R 5 cos(3(y-x)/(y+x))) dx dy = ∫(v = 2 to 10) ∫(u = -v to v) 5 cos(3u/v) * |-1/2| du dv = ∫(v = 2 to 10) (5/2) * (v/3) sin(3u/v) {for u = -v to v} dv = ∫(v = 2 to 10) (5/6) * v (sin 3 - sin(-3)) dv = ∫(v = 2 to 10) (5/6) sin 3 * 2v dv = (5/6) sin 3 * v^2 {for v = 2 to 10} = 80 sin 3." |