Double integral

 

 

Question

        

         Evaluate the integral:

                                                

 

 

 

Analysis

         The following principles or method cannot help:

                 Fundamental theorem of calculus

                 Integration by parts

                 Substitution method

     The integrand of the given integral is an even function.

                         .

 

 

 

Double integral

 

Let

              

Since  x  is a dummy variable, we have

        

 

(1) ´ (2),

        

        

Now, change  (3)  to polar form,

        

         The differential of area =  dA  = dx dy

         The similar differential in polar form can be reasoned like this:

                 Let      dr = r2 – r1

                 Since      dr  is small, we put        r » r1 » r2

                

                

         Since  r2 = x2 + y2,  we get:

 

                

                        

                        

                

                

 

 

 

The study of double integrals is discussed in multivariate calculus.

The change of  dA = dx dy = r dr dq   involves the study of “Jacobian”, which is interesting but a bit involved for secondary students.