14.1 Double Integrals



1. Evaluate the integral  .

  

        

        


2. Evaluate the integral  by changing the order of the integration.

 


3. Evaluate the integral by changing the order of the integration.


4. Evaluate the integral by using the change of variables to the polar coordinates.


5.  Find 

  

        

6. Find .


7. Let . Evaluate .


8. Let . Evaluate .


9. Evaluate , where is the region bounded by , ,

    and .

 

Using the polar coordinate system, the region is represented as follows: and .

 Here we used . Then the given integral becomes

    

     

     


     .


10. Find the volume of the solid with .


11. Evaluate .


12. Prove that

.


13-14. If the density function over the following domain , then find the center of mass, the moments of inertia and .


13.

 



14.