de-CH
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Gebietsintegral
int2-01-01
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19440
randRangeExclude(-8,8,[0,1]) randRangeExclude(-8,8,[0,1,A]) randRange(1,8) randRange(1,8)

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

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



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a \displaystyle \int \int_\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} (negParens(A)\cdot x + negParens(B)\cdot y) dy dx.

Die innere Integration ist

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

Damit erhalten wir für die äussere Integration:

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