2dintegral - 2D INTEGRALS Math21a O Knill HOMEWORK 14.1 18...

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Unformatted text preview: 2D INTEGRALS Math21a, O. Knill HOMEWORK: 14.1: 18, 44 14.2: 24,26 14.3: 26 1D INTEGRATION IN 100 WORDS. If f ( x ) is a continuous function, then integraltext b a f ( x ) dx can be defined as a limit of the Riemann sum f n ( x ) = ∑ x k ∈ [ a,b ] f ( x k )Δ x for n → ∞ with x k = k/n and Δ x = 1 /n . This integral divided by | b- a | is the average of f on [ a, b ]. The integral integraltext b a f ( x ) dx can be interpreted as an signed area under the graph of f , which can be negative too. If f ( x ) = 1, the integral is the length of the interval. The function F ( x ) = integraltext x a f ( y ) dy is called an anti-derivative of f . The fundamental theorem of calculus states F ′ ( x ) = f ( x ). This allows to compute integrals by inverting differentiation. Differentiation rules like the Leibnitz rule become integration rules like integration by part, the chain rule becomes partial integration. Note that unlike the deriva- tive, anti-derivatives can not always be expressed in terms of known functions. An example is: F ( x ) = integraltext x e − t 2 dt . Often, the anti-derivative can be found: Example: f ( x ) = cos 2 ( x ) = (cos(2 x ) + 1) / 2 , F ( x ) = x/ 2- sin(2 x ) / 4....
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