Integrate[
      Sin[m Pi x/a] Sin[n Pi x/a], {x, 0, a}]

w = 
        A11 Sin[ Pi x/a] Sin[Pi y/b] + 
          A31 Sin[ 3  Pi x/a] Sin[Pi y/b] + 
          A13 Sin[ Pi x/a] Sin[3  Pi y/b] + 
          A33 Sin[ 3  Pi x/a] Sin[
              3 Pi y/b];
WA11 = 
      Integrate[
        Integrate[(D[w, {x, 2}] + 
                D[w, {y, 2}])   D[(D[w, {x, 2}] + D[w, {y, 2}]), 
              A11], {y, 0, b}], {x, 0, a}]