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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §5.5 The Definite Integral, #29 · Calculus · Integrals

For … in Problem 28 find the difference … (which approaches zero…

Suppose $M (x)$ is a multistep function with jumps of $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ , at the points $x = \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ . Draw a rough graph with $M (0) = 0$ and $M (1) = 1$ . With $Δ x = \frac{1}{3}$ find $S$ and $s$ .

For $M (x)$ in Problem 28 find the difference $S - s$ (which approaches zero as $Δ x \to 0$ ). What is the area under the graph?

This exercise builds on another exercise in its source book, which isn't shown here.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.5 The Definite Integral, #29

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