ObsidianVault/College/AU 26/MATH 224 (Multivar)/Class 10-2.md

#uw/notes #uw/class/math224

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• 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)