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