INTEGRATION OVER NON-RECTANGULAR BOUNDED REGIONS : DOUBLE INTEGRALS #integration

CONTENTS  :

1. Definition of a function on a bounded region.

2. Definition of integration over bounded region.

3. few problems on integration over some non rectangular bounded regions

1. DEFINITION : 

                            Let E be a bounded region in R^2. Let R be the rectangular region enclosing E.

                            We define a function F(x,y) over R as follows :

                            F(x,y) = f(x,y) ,∀ x in E and F(x,y) = 0 ∀ x not in E.

2. DEFINITION :

                          A function f(x,y) over a bounded region E is said to be integrable if F(x,y) is 

                         integrable over R. 

                         ∴      We write ∬ f(x,y) dxdy = ∬ F(x,y)  dxdy.                   

 
FEW PROBLEMS ON INTEGRATION OVER SOME NON- RECTANGULAR BOUNDED REGIONS : 

1. Problem :

 Sketch the region of integration and evaluate ∬ xsiny dydx where x=0  to x=𝛑 and y-0 to y=x

Solution : 

 2. Problem :

Sketch the region of integration and write an equivalent double integral with the order of integration reversed for ∬ 3y dxdy where y=0 to y=1 and x= -√(1-y^2) to y=√(1-y ^2). 

Solution : 

   

3. Problem :

 In the integral ∬ (4-y) dydx, change the order of integration and evaluate the integral where x=2 to x=4 and y=4/x to y=(20-4x)/(8-x).

Solution : 

  

   

4. Problem :

Change the order of integration and hence show that ∬dx dy/[ ( 1+e^y) √(1-x^2-y^2)]   = (𝝿 /2)   log(2e/(1+e) where x-0 to x=1 and y=0 to y=√(1-x^2) .

Solution :

  

     

5. Problem :

Evaluate ∬ f(x,y) dxdy where f(x,y)= (2y-1)/ x+1 , and E is the region bounded by x=0,y=0, y=2x-4. 

Solution : 

  

6. Problem :

Evaluate ∬ √(4x^2-y^2) dxdy where E is the region bounded by the lines y=0,y=x and x=1.

Solution :

 

7. Problem:

 Evaluate ∬xy(x+y) dxdy where E is the region bounded by y=x^2 and y=x.

Solution :

   

8. Problem :

    Evaluate  ∬f(x,y)dxdy , where f(x,y)= X²+y² and E={(x,y)/y=X²,x=2,y=1}.

Solution : 

  

9. Problem :

 Show that  ∫xy² dy- x²y dx = 35 a⁴π/16 where C is the counter clockwise curve of the cardiod 

       r=a(1+cosθ ).

Solution :

   

   

   

10. Problem :

  Evaluate ∬xy(x+y)² /(x²+y²) dxdy where E is region bounded y=0,y=x,x²+y²=a²                        in the first quadrant.

Solution :

   

    

   

11. Problem :

 Evaluate ∬ e^(x²+y²) dxdy , where E is the semi circular region bounded by the X-axis and                               the curve y=√(1-x²).

Solution :

 

   

12. Problem :

 Using polar coordinates, show that ∫dx ∫√X²+y²    dy =1/6[√2 + log ( 1+√2) ].

Solution :