# 4.2 The Definite Integral - The Definite Integral Section...

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The Definite IntegralSection 4.2
Defn. of a Riemann SumLet fbe defined on the closed interval [a, b], and let be a partition of [a, b] given bya = x0 < x1 < x2 < . . .< xn-1 < xn= b,Where is the length of the ith subinterval. If ci is any point in the ith subinterval, then the sumix
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Defn. of a Definite IntegralIf fis defined on the closed interval [a, b] and the limitexists, then fis integrable on [a, b], and the limit is denoted by 01lim()niiif cx∆ →=01lim()( )bniiiaf cxf x dx∆ →==
Continuity Implies IntegrabilityIf a function fis continuous on the closed interval [a, b], then fis integrable on [a, b].
Thm. 4.5 The Definite Integral as the Area of a RegionIf fis continuous and nonnegative on the