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### Section 4-4 : Double Integrals in Polar Coordinates

1. Evaluate $$\displaystyle \iint\limits_{D}{{3x{y^2} - 2\,dA}}$$ where $$D$$ is the unit circle centered at the origin.
2. Evaluate $$\displaystyle \iint\limits_{D}{{4x - 2y\,dA}}$$ where $$D$$ is the top half of region between $${x^2} + {y^2} = 4$$ and $${x^2} + {y^2} = 25$$.
3. Evaluate $$\displaystyle \iint\limits_{D}{{6xy + 4{x^2}\,dA}}$$ where $$D$$ is the portion of $${x^2} + {y^2} = 9$$ in the 2nd quadrant.
4. Evaluate $$\displaystyle \iint\limits_{D}{{\sin \left( {3{x^2} + 3{y^2}} \right)\,dA}}$$ where $$D$$ is the region between $${x^2} + {y^2} = 1$$ and $${x^2} + {y^2} = 7$$.
5. Evaluate $$\displaystyle \iint\limits_{D}{{{{\bf{e}}^{1 - {x^{\,2}} - {y^{\,2}}}}\,dA}}$$ where $$D$$ is the region in the 4th quadrant between $${x^2} + {y^2} = 16$$ and $${x^2} + {y^2} = 36$$.
6. Use a double integral to determine the area of the region that is inside $$r = 6 - 4\cos \theta$$.
7. Use a double integral to determine the area of the region that is inside $$r = 4$$ and outside $$r = 8 + 6\sin \theta$$.
8. Evaluate the following integral by first converting to an integral in polar coordinates. $\int_{{ - 2}}^{0}{{\int_{{ - \sqrt {4 - {y^{\,2}}} }}^{{\sqrt {4 - {y^{\,2}}} }}{{\,\,\,{x^2}\,dx}}\,dy}}$
9. Evaluate the following integral by first converting to an integral in polar coordinates. $\int_{{ - 1}}^{1}{{\int_{0}^{{\sqrt {1 - {x^{\,2}}} }}{{\,\,\,\sqrt {{x^2} + {y^2}} \,dy}}\,dx}}$
10. Use a double integral to determine the volume of the solid that is below $$z = 9 - 4{x^2} - 4{y^2}$$ and above the $$xy$$-plane.
11. Use a double integral to determine the volume of the solid that is bounded by $$z = 12 - 3{x^2} - 3{y^2}$$ and $$z = {x^2} + {y^2} - 8$$.
12. Use a double integral to determine the volume of the solid that is inside both the cylinder $${x^2} + {y^2} = 9$$ and the sphere $${x^2} + {y^2} + {z^2} = 16$$.
13. Use a double integral to derive the area of a circle of radius $$a$$.
14. Use a double integral to derive the area of the region between circles of radius a and b with $$\alpha \le \theta \le \beta$$. See the image below for a sketch of the region.