Paul's Online Notes
Home / Calculus III / Multiple Integrals / Double Integrals in Polar Coordinates
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 4-4 : 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