Change of Variables :

1. Evaluate \(\displaystyle{ \iint_R (x-3y)  \, dA} \), where \(R\) is the triangular region with vertices \( (0, 0)\), \( (2, 1)\) and \( (1, 2)\). Use \(x = 2u+v, \, y=u+2v \).   solution

2. Evaluate \(\displaystyle{ \iint_E x^2 \, dA} \), where \(E\) is the region bounded by the ellipse \(9x^2+4y^2=36\). Use \(x = 2u, \, y=3v \).  solution

3. Use the Change of Variables Theorem to compute the following double integral using the given substitutions where \( R\) is the parallelogram in the \(xy\)-plane with vertices \((-1,3)\), \( (1,-3)\), \((3,-1)\), and \((1,5)\). solution
\[ \iint_R (4x+8y)\, dA, \: \: x=\frac{1}{4}(u+v), \quad \, y=\frac{1}{4}(v-3u)\]  

4. Evaluate  \(\displaystyle{ \iint_R (4x^2-y^2)^2 \, dA} \) by appropriate change of variables where \(R\) is the region bounded by the lines \(x=0, \: y=2x \), and  \(y=2-2x\).    solution

5. Evaluate \(\displaystyle{ \iint_R e^{x+y} \, dA} \) by the following transformation \(y-x=u, \: x+y=v\), where \(R\) is the region bounded by the lines: \(y=x, \, y=-x+2, \, y=x+2\), and \(y=-x\).  solution

6. Evaluate  \(\displaystyle{ \iint_T \cos\left( \frac{3(y-x)}{y+x} \right) \, dA} \) by appropriate change of variables where \(T\) is the trapezoidal region with vertices \( (2,0)\), \( (5, 0)\), \( (0, 5)\) and \( (0, 2)\).   solution

7. Evaluate \(\displaystyle{ \iint_R (x-y) \, dy\, dx} \), where \(R\) is the parallelogram joining the points (1,2), (3,4), (4,3), and (6,5) by making appropriate change of variables. solution

8. (not in Final) Evaluate the integral using the given change of variables \( \displaystyle{\iint_R (x^2-xy+y^2)\, dA} \) where \(R\) is the region bounded by the ellipse \( x^2-xy+y^2=2; \: x=\sqrt{2}\, u -\sqrt{2/3}\, v, \: y=\sqrt{2}\, u +\sqrt{2/3} \, v\).   solution