   Chapter 4.R, Problem 2CC

Chapter
Section
Textbook Problem

# (a) Write the definition of the definite integral of a continuous function from a to b.(b) What is the geometric interpretation of ∫ a b f ( x )   d x if f ( x ) ≥ 0 ?(c) What is the geometric interpretation of ∫ a b f ( x )   d x if f ( x ) takes on both positive and negative values? Illustrate with a diagram.

To determine

(a)

To write:

The definition of the definite integral of a continuous function from a to b

Explanation

If f is a function defined for axb, we divide the interval [a, b] into n subintervals of equal width x=b-an. let x0=a, x1, x2, .., xn(=b) be the endpoints of these subintervals and we let x1*, x2*, .., xn* be any sample points in these subintervals, so xi* lies in ith subinterval [xi-1, xi]

To determine

(b)

To explain:

The geometric interpretation of abf(x)dx if fx0

To determine

(c)

To explain:

The geometric interpretation of abf(x)dx if fx takes  both positive and negative values and illustrate it with the diagram

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