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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §5.5 The Definite Integral, #25 · Calculus · Integrals

If … is increasing for … the difference … is the area…

If $v (x)$ is increasing for $a ⩽ x ⩽ b$ , the difference $S - s$ is the area of the rectangle minus the area of the rectangle. Those areas approach zero. So every increasing function on $[a, b]$ is Riemann integrable.

If $v (x)$ is increasing for $a \leq x \leq b,$ the difference $S - s$ is the area of the ____ rectangle minus the area of the ____ rectangle. Those areas approach zero. So every increasing function on $[a, b]$ is Riemann integrable.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.5 The Definite Integral, #25

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