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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(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 = 0.5*A*Y*X*X+B*Y*Y*X*0.5

Es ist \displaystyle \int\int_{\orange{D}} f(x,y) dA = \int_0^{X} \int_0^{Y} (Ax + By) \, dy dx.

Die innere Integration ist

\displaystyle \int_0^{Y} (Ax + B y) \, dy = (A x) \cdot y + fractionReduce(B,2)\cdot y^2\bigg|_0^{Y} = A*Yx + fractionReduce(B*Y*Y,2).

Damit erhalten wir für die äussere Integration:

\displaystyle \int_0^{X}\left(A*Yx + fractionReduce(B*Y*Y,2) \right) dx = fractionReduce(A*Y,2) x^2 + fractionReduce(B*Y*Y,2)x \bigg|_0^{X} = fractionReduce(A*Y*X*X+B*Y*Y*X,2).