Question

For the double integral `I=int_0^6 int_y^6 xy sqrt(1+x^4)dxdy`:

  1. sketch the region of integration;
  2. reverse the order of integration and then evaluate the integral analytically.

Answer

  1. reverse order of integrationRegion `D` is shown.

  2. `I=int_0^6int_y^6xysqrt(1+x^4)dxdy=`

    `=int_0^6int_0^x xysqrt(1+x^4)dydx=`

    `=int_0^6xsqrt(1+x^4)int_0^xydydx=`

    `=int_0^6xsqrt(1+x^4)y^2/2|_0^xdx=`

    `=1/2int_0^6x^3sqrt(1+x^4)dx=1/12(1+x^4)^(3/2)|_0^6=`

    `=1/12(1+6^4)^(3/2)-1/12(1+0^4)^(3/2)=`

    `=1/12(1297sqrt(1297)-1)`.