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### Section 15.2 : Iterated Integrals

1. Compute the following double integral over the indicated rectangle (a) by integrating with respect to x first and (b) by integrating with respect to y first. $\iint\limits_{R}{{12x - 18y\,dA}}\hspace{0.25in}\hspace{0.25in}R = \left[ { - 1,4} \right] \times \left[ {2,3} \right]$ Solution

For problems 2 – 8 compute the given double integral over the indicated rectangle.

1. $$\displaystyle \iint\limits_{R}{{6y\sqrt x - 2{y^3}\,dA}}\hspace{0.25in}\hspace{0.25in}R = \left[ {1,4} \right] \times \left[ {0,3} \right]$$ Solution
2. $$\displaystyle \iint\limits_{R}{{\frac{{{{\bf{e}}^x}}}{{2y}} - \frac{{4x - 1}}{{{y^2}}}\,dA}}\hspace{0.25in}\hspace{0.25in}R = \left[ { - 1,0} \right] \times \left[ {1,2} \right]$$ Solution
3. $$\displaystyle \iint\limits_{R}{{\sin \left( {2x} \right) - \frac{1}{{1 + 6y}}\,dA}}\hspace{0.25in}R = \left[ {\frac{\pi }{4},\frac{\pi }{2}} \right] \times \left[ {0,1} \right]$$ Solution
4. $$\displaystyle \iint\limits_{R}{{y{{\bf{e}}^{{y^{\,2}} - 4x}}\,dA}}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}R = \left[ {0,2} \right] \times \left[ {0,\sqrt 8 } \right]$$ Solution
5. $$\displaystyle \iint\limits_{R}{{x{y^2}\,\sqrt {{x^2} + {y^3}} \,dA}}\hspace{0.25in}\hspace{0.25in}R = \left[ {0,3} \right] \times \left[ {0,2} \right]$$ Solution
6. $$\displaystyle \iint\limits_{R}{{xy\cos \left( {y\,{x^2}} \right)\,dA}}\hspace{0.25in}\hspace{0.25in}R = \left[ { - 2,3} \right] \times \left[ { - 1,1} \right]$$ Solution
7. $$\displaystyle \iint\limits_{R}{{xy\cos \left( y \right) - {x^2}\,dA}}\hspace{0.25in}\hspace{0.25in}R = \left[ {1,2} \right] \times \left[ {\frac{\pi }{2},\pi } \right]$$ Solution
8. Determine the volume that lies under $$f\left( {x,y} \right) = 9{x^2} + 4xy + 4$$ and above the rectangle given by $$\left[ { - 1,1} \right] \times \left[ {0,2} \right]$$ in the $$xy$$-plane. Solution