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.

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.

Let \(\mathrm{f}(\mathrm{x})=1, \mathrm{x}\) rational and \(=0, \mathrm{x}\) rational be defined in the interval \([a, b]\). Show that is not Riemann integrable

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.

Given that a bounded function \(\mathrm{f}(\mathrm{x})\) is Riemann integrable in \([a, b]\) if and only if given any \(\varepsilon>0\) there exists a partition with upper and lower sums \(\mathrm{U}\) and \(\mathrm{L}\) such that \(\mathrm{U}-\mathrm{L}<\varepsilon\), prove that a continuous function \(\mathrm{f}(\mathrm{x})\) in \([\mathrm{a}, \mathrm{b}]\) is Riemann integrable in \([\mathrm{a}, \mathrm{b}]\).

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

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 $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{c} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}+\mathrm{b} \int_{\mathrm{c}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} $$

a) Prove that if \(\mathrm{f}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then \(\mid \mathrm{f}(\mathrm{x})\) is Riemann integrable on the same interval. b) Prove that if \(\mathrm{f}(\mathrm{x})\) is Riemann integrable on \([\mathrm{a}, \mathrm{b}]\) then $$ \mathrm{b} \int_{\mathrm{a}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\left|\leq \mathrm{b} \int_{\mathrm{a}}\right| \mathrm{f}(\mathrm{x}) \mid \mathrm{d} \mathrm{x} $$

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.

Let \(\quad F(y)=\pi \int_{0} \sin (x y) d x\) Use Leibniz's rule to find \(\mathrm{F}^{\prime}(\mathrm{y})\).

Find \(\mathrm{F}^{\prime}(\mathrm{x})\) where a) \(F(x)=x \int_{0} e^{-(x) 2(t) 2} d t\) b) \(F(x)=\sin x \int_{(x) 2}\left(x^{2}-t^{2}\right)^{n} d t\)

Proceed from the defintion of the Stieltjes integral to show that the function f given by $$ \begin{array}{cc} \mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})=0 & 0 \leq \mathrm{x} \leq 1 \\ \text { and }=1 & 1<\mathrm{x} \leq 2 \end{array} $$ is not Stieltjes integrable with respect to \(\mathrm{g}\).

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

Let \(\mathrm{f}\) be a function from \([\mathrm{a}, \mathrm{b}]\) into \(\mathrm{R}\) which is continuous at \(c \in[a, b]\) and let \(X_{c}\) be the characteristic function of \(c\), i.e., $$ \begin{array}{cc} X_{c}(x)=1 & x=c \\ \text { and }=0 & x \neq c . \end{array} $$ Show, using the definition of the Stieltjes integral that \(b_{a} f d X_{c}=0 \quad c \in(a, b)\) $$ \begin{array}{ll} \text { and }=-\mathrm{f}(\mathrm{a}) & \mathrm{c}=\mathrm{a} \\ \text { and }=\mathrm{f}(\mathrm{b}) & \mathrm{c}=\mathrm{b} \text { . } \end{array} $$

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 \(\rho(x)\).