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### Section 15.3 : Double Integrals over General Regions

1. Evaluate$$\displaystyle \iint\limits_{D}{{42{y^2} - 12x\,dA}}$$ where $$D = \left\{ {\left( {x,y} \right)|0 \le x \le 4,{{\left( {x - 2} \right)}^2} \le y \le 6} \right\}$$ Solution
2. Evaluate $$\displaystyle \iint\limits_{D}{{2y{x^2} + 9{y^3}\,dA}}$$ where $$D$$ is the region bounded by $$\displaystyle y = \frac{2}{3}x$$ and $$y = 2\sqrt x$$. Solution
3. Evaluate $$\displaystyle \iint\limits_{D}{{10{x^2}{y^3} - 6\,dA}}$$ where $$D$$ is the region bounded by $$x = - 2{y^2}$$ and $$x = {y^3}$$. Solution
4. Evaluate $$\displaystyle \iint\limits_{D}{{x\left( {y - 1} \right)\,dA}}$$ where $$D$$ is the region bounded by $$y = 1 - {x^2}$$ and $$y = {x^2} - 3$$. Solution
5. Evaluate $$\displaystyle \iint\limits_{D}{{5{x^3}\cos \left( {{y^3}} \right)\,dA}}$$ where $$D$$ is the region bounded by $$y = 2$$, $$\displaystyle y = \frac{1}{4}{x^2}$$ and the $$y$$-axis. Solution
6. Evaluate $$\displaystyle \iint\limits_{D}{{\frac{1}{{{y^{\frac{1}{3}}}\left( {{x^3} + 1} \right)}}\,dA}}$$ where $$D$$ is the region bounded by $$x = - {y^{\frac{1}{3}}}$$, $$x = 3$$ and the $$x$$-axis. Solution
7. Evaluate $$\displaystyle \iint\limits_{D}{{3 - 6xy\,dA}}$$ where $$D$$ is the region shown below. Solution
8. Evaluate $$\iint\limits_{D}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}}$$ where $$D$$ is the region shown below. Solution
9. Evaluate $$\displaystyle \iint\limits_{D}{{7{x^2} + 14y\,dA}}$$ where $$D$$ is the region bounded by $$x = 2{y^2}$$ and $$x = 8$$ in the order given below.
1. Integrate with respect to $$x$$ first and then $$y$$.
2. Integrate with respect to $$y$$ first and then $$x$$.
Solution

For problems 10 & 11 evaluate the given integral by first reversing the order of integration.

1. $$\displaystyle \int_{0}^{3}{{\int_{{2x}}^{6}{{\sqrt {{y^2} + 2} \,dy}}\,dx}}$$ Solution
2. $$\displaystyle \int_{0}^{1}{{\int_{{ - \sqrt y }}^{{{y^{\,2}}}}{{6x - y\,dx}}\,dy}}$$ Solution
3. Use a double integral to determine the area of the region bounded by $$y = 1 - {x^2}$$ and $$y = {x^2} - 3$$. Solution
4. Use a double integral to determine the volume of the region that is between the $$xy$$‑plane and$$f\left( {x,y} \right) = 2 + \cos \left( {{x^2}} \right)$$ and is above the triangle with vertices $$\left( {0,0} \right)$$, $$\left( {6,0} \right)$$ and $$\left( {6,2} \right)$$. Solution
5. Use a double integral to determine the volume of the region bounded by $$z = 6 - 5{x^2}$$ and the planes $$y = 2x$$, $$y = 2$$,$$x = 0$$ and the $$xy$$-plane. Solution
6. Use a double integral to determine the volume of the region formed by the intersection of the two cylinders $${x^2} + {y^2} = 4$$ and $${x^2} + {z^2} = 4$$. Solution