I need to evaluate the following integral: ∫ ∫ G ln ⁡ ( x 2 + y 2 ) x 2 + y 2 d x d y

Question

I need to evaluate the following integral:  ∫      ∫    G              ln      ⁡              (                  x          2                +                  y          2                )                            x        2            +              y        2              d  x  d  y

Kennedy Evans · Accepted Answer

Whenever you transform coordinates, say   x  =  f  (  r  ,  θ  ) and   y  =  g  (  r  ,  θ  ), then  d  x  d  y  =  |  J  |  d  r  d  θwhere  J  =      [                                                                      ∂                x                                            ∂                r                                                                                                        ∂                x                                            ∂                θ                                                                                                                    ∂                y                                            ∂                r                                                                                                        ∂                y                                            ∂                θ                                                          ]  In your case, you will find that   |  J  |  =  r and hence   d  x  d  y  =  r  d  r  d  θ. Hence, your integral becomes      ∫    1    e        ∫    0          2      π                          ln        ⁡        (                  r          2                )                    r        2              r  d  r  d  θ  =      ∫    1    e        ∫    0          2      π                          2        ln        ⁡        (        r        )            r        d  r  d  θ  =  4  π      ∫    1    e                      ln        ⁡        (        r        )            r        d  r  =  4  π                                                              log              2                        ⁡            (            r            )                    2                    |              1        e    =  2  π

Kale Sampson · Accepted Answer

Change to polar coordinates:   x  =  r  cos  ⁡      θ  ,   y  =  sin  ⁡      θ  . Then an area element is   d  x    d  y  =  r    d  r    d  θ and the integral becomes                              ∫          0                      2            π                          d        θ                          ∫          1          e                d        r                r                              log            ⁡                                          r                2                                                          r            2                                              =        2        π                  1          2                          ∫          1                                    e              2                                      d        x                              log            ⁡                          x                                x                                                  =        π                  1          2                [        (        log        ⁡                  x                          )          2                          ]          1                                    e              2                                                                        =        2        π

I need to evaluate the following integral: int int_G (ln(x^2+y^2))/(x^2+y^2)dxdy

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