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

Warm-up-exercises

### Exercise

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

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

### Exercise

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

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

### Exercise

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

### Exercise

Let

${\displaystyle f\colon [a,b]\longrightarrow [c,d]}$

be a staircase function and let

${\displaystyle g\colon [c,d]\longrightarrow \mathbb {R} }$

be a function. Prove that the composite ${\displaystyle {}g\circ f}$ is also a staircase function.

### Exercise

Give an example of a continuous function

${\displaystyle f\colon [a,b]\longrightarrow [c,d]}$

and a staircase function

${\displaystyle g\colon [c,d]\longrightarrow \mathbb {R} }$

such that the composite ${\displaystyle {}g\circ f}$ is not a staircase function.

### Exercise

Determine the definite integral

${\displaystyle \int _{0}^{1}tdt}$

explicitly with upper and lower staircase functions.

### Exercise

Determine the definite integral

${\displaystyle \int _{1}^{2}t^{3}dt}$

explicitly with upper and lower staircase functions.

### Exercise

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

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

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

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

### Exercise

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

${\displaystyle {}a=a_{0}

such that the restrictions

${\displaystyle {}f_{i}=f{|}_{[a_{i-1},a_{i}]}\,}$

are Riemann-integrable.

### Exercise

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

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

### Exercise

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

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

### Exercise

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

Hand-in-exercises

### Exercise

Let

${\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }$

be two staircase functions. Prove that ${\displaystyle {}f+g}$ is also a staircase function.

### Exercise

Determine the definite integral

${\displaystyle \int _{a}^{b}t^{2}dt}$

as a function of ${\displaystyle {}a}$ and ${\displaystyle {}b}$ explicitly with lower and upper staircase functions.

### Exercise

Determine the definite integral

${\displaystyle \int _{-2}^{7}-t^{3}+3t^{2}-2t+5dt}$

explicitly with upper and lower staircase functions.

### Exercise

Prove that for the function

${\displaystyle ]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},}$

neither the lower nor the upper integral exist.

### Exercise

Prove that for the function

${\displaystyle ]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{\sqrt {x}}},}$

the lower integral exists, but the upper integral does not exist.

### Exercise

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

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

be a monotone function. Prove that ${\displaystyle {}f}$ is Riemann-integrable.