Chapter 4.R, Problem 2CC

(a) Write the definition of the definite integral of a continuous function from a to b.

(b) What is the geometric interpretation of ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ if $f\left(x\right)\ge 0$?

(c) What is the geometric interpretation of ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ if $f\left(x\right)$ takes on both positive and negative values? Illustrate with a diagram.