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College/AU 26/MATH 224 (Multivar)/Chapter 15.1 - Double Integrals over Rectangles.md

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+#uw/notes #uw/class/math224
+- - -
+- definite integral represents the signed area between a function and the x axis between the two bounds
+	- $\int_{a}^b f(x) dx$
+	- for functions with one variable
+	- riemann sums: $$\int_{a}^b f(x) dx = \lim_{ n \to \infty } f(x^*_{k}) \Delta x$$
+- double integral
+	- $f(x,y)$
+	- $$\int \int_{R} f(x,y) dA$$
+	- represents the signed volume between the xy plane and the graph of $z=f(x,y)$ above the region $R$
+	- defining region
+		- $$R=[a,b] \times [c,d]$$
+		- x values are between $a$ and $b$ , y values are between $c$ and $d$
+	- slice up solid into equations that can be integrated with one variable
+	- $$\int \int_{[a,b] \times [c,d]} f(x,y) dA = \int^b_{a} \int^d_{c}f(x,y)dy dx$$
+	- Fubini's theorem
+		- $$\int_{a}^b \int_{c}^d f(x,y) dy dx = \int_{c}^d \int_{a}^b f(x,y) dx dy$$
+		- can integrate either way (x or y first)