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.

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 \(\rho(x)\).

Suppose f is defined on \([0,2]\) as follows: $$ \begin{aligned} &\mathrm{f}(\mathrm{x})=1 \text { for } 0 \leq \mathrm{x}<1 \\ &\text { and }=2 \text { for } 1 \leq \mathrm{x} \leq 2 \text { . } \end{aligned} $$ Show that \(\mathrm{f}\) is Riemann integrable.

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.

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 $\mathrm{b} \int_{\mathrm{a}}\left[\mathrm{f}_{1}(\mathrm{x})+\mathrm{f}_{2}(\mathrm{x})\right] \mathrm{d} \mathrm{x}=\mathrm{b} \int_{\mathrm{a}} \mathrm{f}_{1}(\mathrm{x}) \mathrm{dx}+\mathrm{b} \int_{\mathrm{a}} \mathrm{f}_{2}(\mathrm{x}) \mathrm{dx}$

Evaluate the Stieltjes integral $^{1} \int_{-1} \mathrm{x} \mathrm{d}|\mathrm{x}|$.