ObsidianVault/Running Start/MATH&163 - Calculus 3/Class 11-20.md

#rs/notes #rs/class/math163

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• 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