Unformatted text preview: the beginning of the interval, namely, Δ xf ( x ). Alternately, the integral may be interpreted as the area under the curve between x and x + Δ x . When Δ x is very small, this will be very close to the area of the rectangle with base Δ x and height f ( x ); again this is Δ xf ( x ). If we accept this, we may proceed: lim Δ x → 1 Δ x Z x +Δ x x f ( t ) dt = lim Δ x → Δ xf ( x ) Δ x = f ( x ) , which is what we wanted to show. It is still true that we are depending on an interpretation of the integral to justify the argument, but we have isolated this part of the argument into two facts that are not too...
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 Spring '07
 JonathanRogawski
 Math, Calculus, Derivative, lim, dt, ∆x

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