12 lines
842 B
Markdown
12 lines
842 B
Markdown
#rs/notes #rs/class/math163
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- Double integrals
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- used to find volume underneath 3D curve $$\int \int f(x, y)dA = \lim_{ m,n \to \infty } \sum_{i=1}^m \sum_{j=1}^n f(x_{i}^*, y_{j}^*)\Delta A$$
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- Properties
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- sum: $$\int \int [f(x, y) + g(x, y)]dA = \int \int f(x, y)dA + \int \int g(x, y)dA$$
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- constant: $$\int \int cf(x, y)dA = c\int \int f(x, y)dA$$
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- Iterated integrals
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- the iterated integral for a function $f(x, y)$ over the rectangular region $R = [a, b] \times [c,d]$ is $$\int_{a}^b \int_{c}^d f(x, y)dy \ dx = \int_{a}^b\left[ \int_{c}^d f(x, y) dy\right]dx$$
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- Fubini's theorem
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- if a function is continuous over the region, then the double integral equals the iterated integral: $$\int \int f(x, y) dA = \int \int f(x, y)dx \ dy = \int_{a}^b \int_{c}^d f(x, y) dx \ dy = \int_{c}^d \int_{a}^b f(x, y) dy \ dx$$
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