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Gebietsintegral über Rechteck
int2-01-01
multiple
10192
randRangeExclude(-8,8,[0,1,-1]) randRangeExclude(-8,8,[0,1,-1,A]) randRange(-8,8) randRangeExclude(P1x+1,8,[-P1x]) randRange(-8,8) randRangeExclude(P1y+1,8,[-P1y]) randRange(1,8) randRangeExclude(1,8,[X])

Gegeben sei die Funktion f: \mathbb R^2 \to \mathbb R mit f(x,y) = A x + B y.

Berechnen Sie das Integral \displaystyle \int \int_{{\color{orange}D}} f(x,y) dA über dem Gebiet \color{orange}D.



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a \displaystyle \int \int_{{\color{orange}D}} f(x,y) dA = (A*(P2y-P1y)*(P2x*P2x-P1x*P1x) + B*(P2y*P2y -P1y*P1y)*(P2x-P1x))/2

Es ist \displaystyle \int\int_{\orange{D}} f(x,y) dA = \int_{P1x}^{P2x} \int_{P1y}^{P2y} (Ax + By) \, dy dx.

Die innere Integration ist

\displaystyle \int_{P1y}^{P2y} (Ax + B y) \, dy = (A x) \cdot y + fractionReduce(B,2)\cdot y^2\bigg|_{P1y}^{P2y} = A*(P2y-P1y)x + fractionReduce(B*(P2y*P2y -P1y*P1y) ,2).

Damit erhalten wir für die äussere Integration:

\displaystyle \int_{P1x}^{P2x} \left(A*(P2y-P1y)x + fractionReduce(B*(P2y*P2y -P1y*P1y) ,2)\right) dx = fractionReduce(A*(P2y-P1y),2) x^2 + fractionReduce(B*(P2y*P2y -P1y*P1y),2)x \bigg|_{P1x}^{P2x} = fractionReduce((A*(P2y-P1y)*(P2x*P2x-P1x*P1x) + B*(P2y*P2y -P1y*P1y)*(P2x-P1x)) ,2).