Chapter 4.R, Problem 2E

(a) Evaluate the Riemann sum for

$f\left(x\right)=x^2-x0\le x\le 2$

With four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

(b) Use the definition of a definite integral (with right endpoints) to calculate the value of the integral

${\displaystyle {\int }_0^2\left(x^2-x\right)}dx$

(c) Use the Fundamental Theorem to check your answer to part (b).

(d) Draw a diagram to explain the geometric meaning of the integral in part (b).