Loading Solution   
Paul's Online Math Notes
Calculus III (Practice Problems) / Multiple Integrals / Double Integrals over General Regions   [Notes] [Practice Problems] [Assignment Problems]

Calculus III - Practice Problems
Applications of Partial Derivatives Previous Chapter   Next Chapter Line Integrals
Iterated Integrals Previous Section   Next Section Double Integrals in Polar Coordinates

 

1. Evaluate  where   [Solution]

 

2. Evaluate  where D is the region bounded by  and . [Solution]

 

3. Evaluate  where D is the region bounded by  and . [Solution]

 

4. Evaluate  where D is the region bounded by  and . [Solution]

 

5. Evaluate  where D is the region bounded by ,  and the y-axis. [Solution]

 

6. Evaluate  where D is the region bounded by ,  and the x-axis. [Solution]

 

7. Evaluate  where D is the region shown below.

[Solution]

 

8. Evaluate  where D is the region shown below.

[Solution]

 

9. Evaluate  where D is the region bounded by  and  in the order given below.

   (a) Integrate with respect to x first and then y.

   (b) Integrate with respect to y first and then x.

[Solution]

 

For problems 10 & 11 evaluate the given integral by first reversing the order of integration.

 

10.    [Solution]

 

11.    [Solution]

 

12. Use a double integral to determine the area of the region bounded by  and . [Solution]

 

13. Use a double integral to determine the volume of the region that is between the xy-plane and  and is above the triangle with vertices ,  and . [Solution]

 

14. Use a double integral to determine the volume of the region bounded by  and the planes , ,  and the xy-plane. [Solution]

 

15. Use a double integral to determine the volume of the region formed by the intersection of the two cylinders  and . [Solution]

 

Problem Pane
Iterated Integrals Previous Section   Next Section Double Integrals in Polar Coordinates
Applications of Partial Derivatives Previous Chapter   Next Chapter Line Integrals

Calculus III (Practice Problems) / Multiple Integrals / Double Integrals over General Regions    [Notes] [Practice Problems] [Assignment Problems]

© 2003 - 2017 Paul Dawkins