# Slide 4.5.pptx - The Definite Integral Riemann Sum...

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The Definite Integral
Riemann SumLet’s pick some wkin each subinterval, then build the rectangle with height = f(wk).If f(wk) > 0, rectangle lies above x-axis.If f(wk) < 0, rectangle lies below x-axis.Let f(x) be defined on [a,b] and P be any decomposition of [a,b] into subintervals of the form [xk-1,xk]Riemann sum of f(x) for P is
TheoremIf f is integrable and f(x) > 0 for every x in [a,b] (where a < b), then the area A of the region under the graph of f from a to b is The Definite IntegralDefinitionThe Definite Integral of f from a to b is provided the limit exists where ||p|| is the norm of the partition (largest Δx