diff --git a/College/AU 26/MATH 224 (Multivar)/Class 10-5.md b/College/AU 26/MATH 224 (Multivar)/Class 10-5.md index e3250e0..70db885 100644 --- a/College/AU 26/MATH 224 (Multivar)/Class 10-5.md +++ b/College/AU 26/MATH 224 (Multivar)/Class 10-5.md @@ -17,9 +17,6 @@ $$ Area(x^*)=\int_{x}^{x^2}x^*ydy$$ Since $x$ is bound by $1 [!NOTE] Theorem -> Contents - ### Theorem: If the region on the $XY$ plane for the region being integrated over of $f(x,y)$ ($D$) is continuous and bound by $x=a$ and $x=b$ in the x-axis and two functions $g_{1}(x)$ and $g_{2}(x)$ on the y-axis then the integral $$\int \int_{D} f(x,y)$$ @@ -27,7 +24,8 @@ can be rewritten as $$\int_{a}^b \int_{g_{1}(x)}^{g_{2}(x)} f(x,y) \ dydx$$ This can also be done if $D$ is bound by functions on the x-axis instead of the y-axis -### Property: -If $D=D_{1} \cup D_{2}$ and $D_{1}$ and $D_{2}$ do not intersect except at their boundaries then -$$\int \int_{D} f(x,y) \ dA = \int \int_{D_{1}} f(x,y) \ dA + \int \int_{D_{2}} f(x,y) \ dA$$ -This can be used to split up integrals similar to area in 2D + +> [!note] Property +> If $D=D_{1} \cup D_{2}$ and $D_{1}$ and $D_{2}$ do not intersect except at their boundaries then +> $$\int \int_{D} f(x,y) \ dA = \int \int_{D_{1}} f(x,y) \ dA + \int \int_{D_{2}} f(x,y) \ dA$$ +> This can be used to split up integrals similar to area in 2D