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Calculus III (Assignment Problems) / Multiple Integrals / Double Integrals in Polar Coordinates   [Notes] [Practice Problems] [Assignment Problems]

Calculus III - Assignment Problems
Applications of Partial Derivatives Previous Chapter   Next Chapter Line Integrals
Double Integrals over General Regions Previous Section   Next Section Triple Integrals

 Double Integrals in Polar Coordinates

 

1. Evaluate  where D is the unit circle centered at the origin. 

 

2. Evaluate  where D is the top half of region between  and  .

 

3. Evaluate  where D is the portion of  in the 2nd quadrant.  

 

4. Evaluate  where D is the region between  and

 

5. Evaluate  where D is the region in the 4th quadrant between  and .   

 

6. Use a double integral to determine the area of the region that is inside .  

 

7. Use a double integral to determine the area of the region that is inside  and outside .  

 

8. Evaluate the following integral by first converting to an integral in polar coordinates.

 

 

 

9. Evaluate the following integral by first converting to an integral in polar coordinates.

 

  

 

10. Use a double integral to determine the volume of the solid that is below  and above the xy-plane.

 

11. Use a double integral to determine the volume of the solid that is bounded by  and .

 

12. Use a double integral to determine the volume of the solid that is inside both the cylinder  and the sphere .

 

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 .  See the image below for a sketch of the region.

 

 

Double Integrals over General Regions Previous Section   Next Section Triple Integrals
Applications of Partial Derivatives Previous Chapter   Next Chapter Line Integrals

Calculus III (Assignment Problems) / Multiple Integrals / Double Integrals in Polar Coordinates    [Notes] [Practice Problems] [Assignment Problems]

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