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.trash/Class 11-20.md

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+[[Calculus 3]]
+- Double integrals
+	- used to find volume underneath 3D curve $$\int \int f(x, y)dA = \lim_{ m,n \to \infty } \sum_{i=1}^m \sum_{j=1}^n f(x_{i}^*, y_{j}^*)\Delta A$$
+	- Properties
+		- sum: $$\int \int [f(x, y) + g(x, y)]dA = \int \int f(x, y)dA + \int \int g(x, y)dA$$
+		- constant: $$\int \int cf(x, y)dA = c\int \int f(x, y)dA$$
+- Iterated integrals
+	- the iterated integral for a function $f(x, y)$ over the rectangular region $R = [a, b] \times [c,d]$ is $$\int_{a}^b \int_{c}^d f(x, y)dy \ dx = \int_{a}^b\left[ \int_{c}^d f(x, y) dy\right]dx$$
+	- Fubini's theorem
+		- if a function is continuous over the region, then the double integral equals the iterated integral: $$\int \int f(x, y) dA = \int \int f(x, y)dx \ dy = \int_{a}^b \int_{c}^d f(x, y) dx \ dy = \int_{c}^d \int_{a}^b f(x, y) dy \ dx$$
+		-