# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 23

Warm-up-exercises

### Exercise

Determine the Riemann sum over ${}[-3,+4]$ of the staircase function

${}f(t)={\begin{cases}5,{\text{ if }}-3\leq t\leq -2\,,\\-3,{\text{ if }}-2 ### Exercise

a) Subdivide the interval ${}[-4,5]$ in six subintervals of equal length.

b) Determine the Riemann sum of the staircase function on ${}[-4,5]$ , which takes alternately the values ${}2$ and ${}-1$ on the subdivision constructed in a).

### Exercise

Give an example of a function ${}f\colon [a,b]\rightarrow \mathbb {R}$ which assumes only finitely many values, but is not a staircase function.

### Exercise

Let

$f\colon [a,b]\longrightarrow [c,d]$ be a staircase function and let

$g\colon [c,d]\longrightarrow \mathbb {R}$ be a function. Prove that the composite ${}g\circ f$ is also a staircase function.

### Exercise

Give an example of a continuous function

$f\colon [a,b]\longrightarrow [c,d]$ and a staircase function

$g\colon [c,d]\longrightarrow \mathbb {R}$ such that the composite ${}g\circ f$ is not a staircase function.

### Exercise

Determine the definite integral

$\int _{0}^{1}tdt$ explicitly with upper and lower staircase functions.

### Exercise

Determine the definite integral

$\int _{1}^{2}t^{3}dt$ explicitly with upper and lower staircase functions.

### Exercise

Let ${}I=[a,b]$ be a compact interval and let

$f\colon I\longrightarrow \mathbb {R}$ be a function. Consider a sequence of staircase functions ${}{\left(s_{n}\right)}_{n\in \mathbb {N} }$ such that ${}s_{n}\leq f$ and a sequence of staircase functions ${}{\left(t_{n}\right)}_{n\in \mathbb {N} }$ such that ${}t_{n}\geq f$ . Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that ${}f$ is Riemann-integrable and that

${}\lim _{n\rightarrow \infty }\int _{a}^{b}s_{n}(x)dx=\int _{a}^{b}f(x)dx=\lim _{n\rightarrow \infty }\int _{a}^{b}t_{n}(x)dx\,.$ ### Exercise

Let ${}I=[a,b]$ be a compact interval. Prove that ${}f$ is Riemann-integrable if and only if there is a partition

${}a=a_{0} such that the restrictions

${}f_{i}=f{|}_{[a_{i-1},a_{i}]}\,$ are Riemann-integrable.

### Exercise

Let ${}I=[a,b]\subseteq \mathbb {R}$ be a compact interval and let ${}f,g\colon I\rightarrow \mathbb {R}$ be two Riemann-integrable functions. Prove the following statements.

1. If ${}m\leq f(t)\leq M$ for all ${}t\in I$ , then
${}m(b-a)\leq \int _{a}^{b}f(t)dt=M(b-a)\,.$ 2. If ${}f(t)\leq g(t)$ for all ${}t\in I$ , then
${}\int _{a}^{b}f(t)dt\leq \int _{a}^{b}g(t)dt\,.$ 3. We have
${}\int _{a}^{b}f(t)+g(t)dt=\int _{a}^{b}f(t)dt+\int _{a}^{b}g(t)dt\,.$ 4. For ${}c\in \mathbb {R}$ we have ${}\int _{a}^{b}cf(t)dt=c\int _{a}^{b}f(t)dt$ .

### Exercise

Let ${}I=[a,b]$ be a compact interval and let ${}f\colon I\rightarrow \mathbb {R}$ be a Riemann-integrable function. Prove that

${}\vert {\int _{a}^{b}f(t)dt}\vert \leq \int _{a}^{b}\vert {f(t)}\vert dt\,.$ ### Exercise

Let ${}I=[a,b]$ be a compact interval and let ${}f,g\colon I\rightarrow \mathbb {R}$ be two Riemann-integrable functions. Prove that ${}fg$ is also Riemann-integrable.

Hand-in-exercises

### Exercise

Let

$f,g\colon [a,b]\longrightarrow \mathbb {R}$ be two staircase functions. Prove that ${}f+g$ is also a staircase function.

### Exercise

Determine the definite integral

$\int _{a}^{b}t^{2}dt$ as a function of ${}a$ and ${}b$ explicitly with lower and upper staircase functions.

### Exercise

Determine the definite integral

$\int _{-2}^{7}-t^{3}+3t^{2}-2t+5dt$ explicitly with upper and lower staircase functions.

### Exercise

Prove that for the function

$]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$ neither the lower nor the upper integral exist.

### Exercise

Prove that for the function

$]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{\sqrt {x}}},$ the lower integral exists, but the upper integral does not exist.

### Exercise

Let ${}I$ be a compact interval and let

$f\colon I\longrightarrow \mathbb {R}$ be a monotone function. Prove that ${}f$ is Riemann-integrable.