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[See Solution] (a) Use the change of variables u=x+y, v=x-y to evaluate ∫_R √x+y sin (x-y) d d x d y where R is the quadrilateral with vertices at

Question: (a) Use the change of variables \(u=x+y, v=x-y\) to evaluate

\[\int_{R} \sqrt{x+y} \sin (x-y) \mathrm{d} d x d y\]

where \(R\) is the quadrilateral with vertices at

\[(0,0),(2,2),\left(2+\frac{\pi}{2}, 2-\frac{\pi}{2}\right),\left(\frac{\pi}{2},-\frac{\pi}{2}\right)\]

(b) Using change of variables

\[x=u+2 v, y=2 v+a\]

evaluate the integral

\[\int_{\Omega}(x-y) d x d y\]

where \(\Omega\) is a parallelogram whose sides are given by the lines:

\[y=x, y=x+a, y=a, y=3 a\]

and \(a>0\) is a fixed real number.

(c) Using cylindrical coordinates evaluate the integral

\[\int_{V} \sqrt{x^{2}+y^{2}+1} \mathrm{~d} d x d y d z\]

where \(V\) is the region in the first octant

\[\left\{(x, y, z) \in \mathbb{R}^{3} \mid x \geq 0, y \geq 0, z \geq 0\right\}\]

bounded by the cylinder \(x^{2}+y^{2}=1\) and the plane \(z=1\)

Price: $2.99
Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document


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