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### Section 15.4 : Double Integrals in Polar Coordinates

1. Evaluate $$\displaystyle \iint\limits_{D}{{{y^2} + 3x\,dA}}$$ where $$D$$ is the region in the 3rd quadrant between $${x^2} + {y^2} = 1$$ and $${x^2} + {y^2} = 9$$. Solution
2. Evaluate $$\displaystyle \iint\limits_{D}{{\sqrt {1 + 4{x^2} + 4{y^2}} \,dA}}$$ where $$D$$ is the bottom half of $${x^2} + {y^2} = 16$$. Solution
3. Evaluate $$\displaystyle \iint\limits_{D}{{4xy - 7\,dA}}$$ where $$D$$ is the portion of $${x^2} + {y^2} = 2$$ in the 1st quadrant. Solution
4. Use a double integral to determine the area of the region that is inside $$r = 4 + 2\sin \theta$$ and outside $$r = 3 - \sin \theta$$. Solution
5. Evaluate the following integral by first converting to an integral in polar coordinates. $\int_{0}^{3}{{\int_{{ - \sqrt {9 - {x^{\,2}}} }}^{0}{{\,\,\,{{\bf{e}}^{{x^{\,2}} + {y^{\,2}}}}\,dy}}\,dx}}$ Solution
6. Use a double integral to determine the volume of the solid that is inside the cylinder $${x^2} + {y^2} = 16$$, below $$z = 2{x^2} + 2{y^2}$$ and above the $$xy$$-plane. Solution
7. Use a double integral to determine the volume of the solid that is bounded by $$z = 8 - {x^2} - {y^2}$$ and $$z = 3{x^2} + 3{y^2} - 4$$. Solution