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Stefan
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I don't really understand the question but you can try this:

Manipulate[Plot[{Sin[x], Sin[x] Boole[x < y]}, {x, 0, Pi}, 
    Filling -> 2 -> Axis, Frame -> True, 
    ImageSize -> 500, Epilog -> Inset[NIntegrate[Sin[x], 
      {x, 0, y}], {y, 1/2 Sin[y]}, {Right, Center}]], {y, 0, Pi} 
]

I don't really understand the question but you can try this:

Manipulate[Plot[{Sin[x], Sin[x] Boole[x < y]}, {x, 0, Pi}, Filling -> 2 -> Axis, Frame -> True, ImageSize -> 500, Epilog -> Inset[NIntegrate[Sin[x], {x, 0, y}], {y, 1/2 Sin[y]}, {Right, Center}]], {y, 0, Pi}]

I don't really understand the question but you can try this:

Manipulate[Plot[{Sin[x], Sin[x] Boole[x < y]}, {x, 0, Pi}, 
    Filling -> 2 -> Axis, Frame -> True, 
    ImageSize -> 500, Epilog -> Inset[NIntegrate[Sin[x], 
      {x, 0, y}], {y, 1/2 Sin[y]}, {Right, Center}]], {y, 0, Pi} 
]
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I don't really understand the question but you can try this:

Manipulate[Plot[{Sin[x], Sin[x] Boole[x < y]}, {x, 0, Pi}, Filling -> 2 -> Axis, Frame -> True, ImageSize -> 500, Epilog -> Inset[NIntegrate[Sin[x], {x, 0, y}], {y, 1/2 Sin[y]}, {Right, Center}]], {y, 0, Pi}]