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Section 4-3 : Double Integrals over General Regions

1. Evaluate $$\displaystyle \iint\limits_{D}{{8y{x^3}\,dA}}$$ where $$D = \left\{ {\left( {x,y} \right)| - 1 \le y \le 2, - 1 \le x \le 1 + {y^2}} \right\}$$
2. Evaluate $$\displaystyle \iint\limits_{D}{{12{x^2}y - {y^2}\,dA}}$$ where $$D = \left\{ {\left( {x,y} \right)| - 2 \le x \le 2, - {x^2} \le y \le {x^2}} \right\}$$
3. Evaluate $$\displaystyle \iint\limits_{D}{{9 - \frac{{6{y^2}}}{{{x^2}}}\,dA}}$$ where $$D$$ is the region in the 1st quadrant bounded by $$y = {x^3}$$ and $$y = 4x$$.
4. Evaluate $$\displaystyle \iint\limits_{D}{{15{x^2} - 6y\,dA}}$$ where $$D$$ is the region bounded by $$x = \frac{1}{2}{y^2}$$ and $$x = 4\sqrt y$$.
5. Evaluate $$\displaystyle \iint\limits_{D}{{6y{{\left( {x + 6} \right)}^2}\,dA}}$$ where $$D$$ is the region bounded by $$x = - {y^2}$$ and $$x = y - 6$$.
6. Evaluate $$\displaystyle \iint\limits_{D}{{{{\bf{e}}^{{y^{\,2}} + 1}}\,dA}}$$ where $$D$$ is the triangle with vertices $$\left( {0,0} \right)$$, $$\left( { - 2,4} \right)$$ and $$\left( {8,4} \right)$$.
7. Evaluate $$\displaystyle \iint\limits_{D}{{7{y^3}{{\bf{e}}^{{x^{\,2}} + 1}}\,dA}}$$ where $$D$$ is the region bounded by$$y = 2\,\,\sqrt[4]{x}$$, $$x = 9$$ and the $$x$$-axis.
8. Evaluate $$\displaystyle \iint\limits_{D}{{{x^5}\sin \left( {{y^4}} \right)\,dA}}$$ where $$D$$ is the region in the 2nd quadrant bounded by $$y = 3{x^2}$$, $$y = 12$$ and the $$y$$-axis.
9. Evaluate $$\displaystyle \iint\limits_{D}{{xy - {y^2}\,dA}}$$ where $$D$$ is the region shown below.
10. Evaluate $$\displaystyle \iint\limits_{D}{{12{x^3} - 3\,dA}}$$ where $$D$$ is the region shown below.
11. Evaluate $$\displaystyle \iint\limits_{D}{{6{y^2} + 10y{x^4}\,dA}}$$ where $$D$$ is the region shown below.
12. Evaluate $$\displaystyle \iint\limits_{D}{{\frac{{{x^3}}}{{{y^2}}}\,dA}}$$ where $$D$$ is the region bounded by $$\displaystyle y = \frac{1}{{{x^2}}}$$, $$x = 1$$ and $$\displaystyle y = \frac{1}{4}$$ in the order given below.
1. Integrate with respect to $$x$$ first and then $$y$$.
2. Integrate with respect to $$y$$ first and then $$x$$.
13. Evaluate $$\displaystyle \iint\limits_{D}{{xy - {y^3}\,dA}}$$ where $$D$$ is the region bounded by $$y = {x^2}$$, $$y = - {x^2}$$ and $$x = 2$$ in the order given below.
1. Integrate with respect to $$x$$ first and then $$y$$.
2. Integrate with respect to $$y$$ first and then $$x$$.

For problems 14 – 16 evaluate the given integral by first reversing the order of integration.

1. $$\displaystyle \int_{0}^{8}{{\int_{{{y^{\frac{1}{3}}}}}^{2}{{\frac{y}{{{x^7} + 1}}\,dx}}\,dy}}$$
2. $$\displaystyle \int_{{ - 4}}^{0}{{\int_{{\sqrt { - x} }}^{2}{{{x^{ - \,\,\frac{2}{3}}}\,\,\sqrt {{y^{\frac{5}{3}}} + 1} \,dy}}\,dx}}$$
3. $$\displaystyle \int_{0}^{2}{{\int_{{ - x}}^{{3x}}{{5{y^2}{x^3} + 2\,dy}}\,dx}}$$
4. Use a double integral to determine the area of the region bounded by $$x = - {y^2}$$ and $$x = y - 6$$.
5. Use a double integral to determine the area of the region bounded by $$y = {x^2} + 1$$ and $$\displaystyle y = \frac{1}{2}{x^2} + 3$$.
6. Use a double integral to determine the volume of the region that is between the $$xy$$‑plane and $$f\left( {x,y} \right) = 2 - x{y^2}$$ and is above the region in the $$xy$$-plane that is bounded by $$y = {x^2}$$ and $$x = 1$$.
7. Use a double integral to determine the volume of the region that is between the $$xy$$‑plane and $$f\left( {x,y} \right) = 1 + {y^5}\,\,\sqrt {{x^4} + 1}$$ and is above the region in the $$xy$$-plane that is bounded by $$y = \sqrt x$$, $$x = 2$$ and the $$x$$-axis.
8. Use a double integral to determine the volume of the region in the first octant that is below the plane given by $$2x + 6y + 4z = 8$$.
9. Use a double integral to determine the volume of the region bounded by $$z = 3 - 2y$$, the surface $$y = 1 - {x^2}$$ and the planes $$y = 0$$ and $$z = 0$$.
10. Use a double integral to determine the volume of the region bounded by the planes $$z = 4 - 2x - 2y$$, $$y = 2x$$, $$y = 0$$ and $$z = 0$$.
11. Use a double integral to determine the formula for the area of a right triangle with base, $$b$$ and height $$h$$.
12. Use a double integral to determine a formula for the figure below.