Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 5 - Integration - 5.2 Definite Integrals - 5.2 Exercises: 6

Answer

They are connected by the limit: $$\int_a^b f(x)dx = \lim_{\Delta\to0}\sum_{k=1}^nf(x_k^*)\Delta x_k.$$

Work Step by Step

The definite integral is the limit when the width $\Delta = \max\{\Delta x_i\}$ of the arbitrary partition of the interval $[a,b]$ tends to zero of the Riemann sum $\sum_{k=1}^nf(x_k^*)\Delta x_k$ for arbitrary choice $x_k^*\in[x_{i},x_{i+1}]$. Thus the Riemann sum approximates the value of the definite integral and it is a better approximation if the partition contains more points and its width is smaller. Precisely: $$\int_a^b f(x)dx = \lim_{\Delta\to0}\sum_{k=1}^nf(x_k^*)\Delta x_k $$
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