DOUBLE INTEGRALS - PROBLEMS WITH SOLUTIONS #double #integral

 Double Integrals : Problems & Solutions :

                                                                          This page having some problems on Double Integrals sollowed by solutions. 

                                                                           

       

1.    1. Evaluate ∬ xy (x²+y²)dx dy over [0,a;0,a].

2.    2. Evaluate  ∬xy (x²+y²)dx dy over [0,a;0,b].

3.     3. Evaluate  ∬ ye^xy over [0,a;0,b].

4.    4. Evaluate   ∬(x-y)/(x+y) dxdy over [0,1;0,1].

5.   5. Show that ⲫ(h)= ∫{ ∫ (x-y)/(x+y)³ dy}dx is not continuous for h=0 where x and y varies from h to 1.

6.    6. Sketch the region of integration and evaluate the integral  ∬(x+y+1)dxdy where x varies from -1 to 0         and y varies from -1 to 1.

7.    7. Evaluate  ∬y²/(1+x²)  dxdy over [-1,1;0,2].

8.    8. Integrate  ∬f(x,y) =y cosxy over the rectangle 0≤x≤π , 0≤y≤1

9.    9. Evaluate the double integral of f(x,y) =xy(x²+y²) over the rectangle R = [0,1;0,1].

     10.   Evaluate   ∬ f(x,y) dxdy where f(x,y) = √x + y-3x²y and R= [0,1;1,3].

       11. Sketch the region of integration and evaluate the integral ∬ x siny dydx where x limits from 0 to π and y limits from 0 to x.

12. Evaluate ∬ xy dxdy over the region E which is bounded by xy=1, y=0, y=x, x=2.

13. Evaluate ∬√(4x²-y²) dxdy over   E where E is the triangle bounded by the lines y=0, y=x, x=1.

14. Evaluate ∬ xy (x+y) dxdy over  E where E is the region bounded by y=x²  and y=x.

15. Evaluate ∬ f(x,y) dxdy over E where f(x,y)= x²+y²  and E={ (x,y)/ y=x² , x=2,y=1} 

       

     SOLUTIONS  : 

 1. Evaluate ∬ xy (x²+y²)dx dy over [0,a;0,a].

SOLUTION :  

2.    2. Evaluate  ∬xy (x²+y²)dx dy over [0,a;0,b].

     SOLUTION :

                 

  1

3.     3. Evaluate  ∬ ye^xy over [0,a;0,b].

      SOLUTION :

                                

4.    4. Evaluate   ∬(x-y)/(x+y) dxdy over [0,1;0,1].

     SOLUTION :

            

5.   5. Show that ⲫ(h)= ∫{ ∫ (x-y)/(x+y)³ dy}dx is not continuous for h=0 where x and y varies from h to 1.

      SOLUTION :

                           

6. 

                              

 6.     6.  Sketch the region of integration and evaluate the integral  ∬(x+y+1)dxdy where x varies from -1 to 0         and y varies from -1 to 1.

       SOLUTION :   

                        

7.    7. Evaluate  ∬y²/(1+x²)  dxdy over [-1,1;0,2].

       SOLUTION :

               

8.    8. Integrate  ∬f(x,y) =y cosxy over the rectangle 0≤x≤π , 0≤y≤1

     SOLUTION :

             

9.    9. Evaluate the double integral of f(x,y) =xy(x²+y²) over the rectangle R = [0,1;0,1].

     SOLUTION :

                          

     10.   Evaluate   ∬ f(x,y) dxdy where f(x,y) = √x + y-3x²y and R= [0,1;1,3].

     SOLUTION :

                      

      11. Sketch the region of integration and evaluate the integral ∬ x siny dydx where x limits from 0 to π 

      and y limits from 0 to x.

 SOLUTION :

                             

12. Evaluate ∬ xy dxdy over the region E which is bounded by xy=1, y=0, y=x, x=2.

 SOLUTION :

                   

13. Evaluate ∬√(4x²-y²) dxdy over   E where E is the triangle bounded by the lines y=0, y=x, x=1.

SOLUTION :

            

14. Evaluate ∬ xy (x+y) dxdy over  E where E is the region bounded by y=x²  and y=x.

SOLUTION :

                   

15. Evaluate ∬ f(x,y) dxdy over E where f(x,y)= x²+y²  and E={ (x,y)/ y=x² , x=2,y=1} 

SOLUTION :