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Problem 190

Prove the following: a) If \(\mathrm{f}(\mathrm{x}) \leq \mathrm{g}(\mathrm{x})\) and \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \leq \mathrm{b}_{\mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x} $$ b) If \(\mathrm{f}(\mathrm{x})\) is bounded and Riemann integrable and if \(\mathrm{c}\) is any point such that \(\mathrm{C} \in[\mathrm{a}, \mathrm{b}]\) then $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{c} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}+\mathrm{b} \int_{\mathrm{c}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} $$

Expert verified

In summary, we proved the following:
a) If \(f(x) \leq g(x)\), and both functions are Riemann integrable on the interval [a, b], then \(\int_a^b f(x) dx \leq \int_a^b g(x) dx\). We used the definitions of upper and lower sums and the properties of Riemann integrable functions to show this inequality.
b) If \(f(x)\) is bounded and Riemann integrable and c is any point in [a, b], then \(\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx\). We demonstrated this by partitioning the interval [a, b] and showing that the sums of the respective partitions give us the integral over the entire interval.

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Chapter 7

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be bounded and defined on the closed interval \([\mathrm{a}, \mathrm{b}]\). Define the Riemann integral using the concept of partitions.

Chapter 7

Given that \(\mathrm{f}_{1}(\mathrm{x})\) and \(\mathrm{f}_{2}(\mathrm{x})\) are Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) and that $\mathrm{I} \geq \mathrm{J}$ (where I and J represent the upper and lower integrals, respectively, of any Riemann integrable function), prove that \(\mathrm{f}_{1}(\mathrm{x})+\mathrm{f}_{2}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) and that $\mathrm{b} \int_{\mathrm{a}}\left[\mathrm{f}_{1}(\mathrm{x})+\mathrm{f}_{2}(\mathrm{x})\right] \mathrm{d} \mathrm{x}=\mathrm{b} \int_{\mathrm{a}} \mathrm{f}_{1}(\mathrm{x}) \mathrm{dx}+\mathrm{b} \int_{\mathrm{a}} \mathrm{f}_{2}(\mathrm{x}) \mathrm{dx}$

Chapter 7

Find the derivatives of a) $\mathrm{x} \int_{1} \mathrm{t}^{2} \mathrm{dt} \quad\( with respect to \)\mathrm{x}$. b) ${ }^{\text {(t) } 2} \int_{1} \sin \left(\mathrm{x}^{2}\right) \mathrm{d} \mathrm{x}$ with respect to t.

Chapter 7

Suppose \(g\) is a continuously differentiable monotonically increasing function on \([a, b]\) and \(f\) is bounded on \([a, b]\). Prove that $$ b \int_{a} f d g=b \int_{a} f(x) g^{\prime}(x) d x $$ Use this result to find the total mass of a linear distribution on \([a, b]\) with a continuous density function \(\rho(x)\).

Chapter 7

Suppose \(\mathrm{f}\) is defined on \([0,2]\) by \(\mathrm{f}(\mathrm{x})=\mathrm{x}\) if \(0 \leq \mathrm{x}<1\), \(\mathrm{f}(\mathrm{x})=\mathrm{x}-1\) if \(1 \leq \mathrm{x} \leq 2\). Show that \(\mathrm{f}\) is integrable.

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