LaTeX automatically starting new lines and indenting

Question

So in the accompanying image, The text "Note: The "variable of integration" is irrelevant" should be on the same line as the formula under it. In the code I have it all on one line, but it is not displaying that way. The formula above it and all the formulas in the middle need to be left aligned. I have tried flalign, and align but flalign just gives me errors and align is not working for some reason.

Here is the code that I want to display all on 1 line: Note: The "variable of integration" is irrelevant. $$\int_{-2}^{5} x^3 dx = \int_{-2}^{5} y^3 dy = \int_{-2}^{5} t^3 dt = ...$$

And I just need to left align some of the formulas with a little bit of space indented to the right to make it show as different from the text.

Thanks

Here is my full code:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\documentclass[fleqn]{article}
\setlength{\mathindent}{0pt}
\begin{document}


\section{1 Limit of a Riemann Sum}Suppose $f$ is defined on [$a,b$] and $L$ is a real number. We write \\ $$\int_{a}^{b} f(x) dx = \lim_{|| P ||\to\ 0} \sum\limits_{k=1}^n f(C_k)\Delta \chi _k = L$$
if for every $\epsilon$ $>$ 0, there is a  $\delta$ $>$ 0 so that if $P$ is any partition of [$a,b$] with $\left\Vert\left P \left\Vert\left $ $<$ $\delta$ and each $C_k$ is a number in the $k$th subinterval of $P$, then

$$\left\vert\left \sum\limits_{k=1}^n f(C_k)\Delta \chi _k - L \right\vert\right \textless \thinspace \thinspace \epsilon$$
\\
\\
 \newline Note: The "variable of integration" is irrelevant. $$\int_{-2}^{5} x^3 dx = \int_{-2}^{5} y^3 dy = \int_{-2}^{5} t^3 dt = ...$$
 \\
 Ex: Do the following exist?
 \\
 \begin{flalign}
 $$\left \int_{1}^{3} \frac{1}{x^2} dx$$ &&
  \\&&
 $$\left \int_{0}^{2} \frac{1}{x^2} dx$$&&
 \end{flalign}
 \\
 \\
 Ex: Evaluate $$\int_{1}^{4} f(x) dx$$ where $f(x)$ = 

 \end{flalign}

 \\
 \begin{cases}
 $$3x^2 + 1, \quad 1 \leq x \leq 3$$
 \\
 $$28, \quad \quad \quad \quad 3$$ $<$ $$x \leq 4$$

 \end{cases}
 \\
 \\
 \\

 You may use the fact that $$\int_{1}^{3} x^2 dx = \frac{26}{3}$$ where $f(x)$ 


 \section{Miscellaneous Trivia}
 Prove: If A $$\subseteq B, then \quad A \cap C \subseteq B \cap C$$
 \\
 \\
 Fact: $C(n,r)$ = \Bigg(\overset{n}{r}\Bigg) = $$\frac{P(n,r)}{r!} = \frac{\frac{n!}{(n-r)!}}{r!} = \frac{n!}{r!(n-r)!}$$
\end{document}

Answer 1

The posted code produces the error

! LaTeX Error: Two \documentclass or \documentstyle commands.

as there are two \documentclass

If you comment out the second one you get the error

! Undefined control sequence.
<argument> \mathindent 

l.6 \setlength{\mathindent}{0pt}

as that length isn't defined

Commenting out that line you get the error

! Missing delimiter (. inserted).
<to be read again> 
                   $
l.11 ... with $\left\Vert\left P \left\Vert\left $
                                                   $<$ $\delta$ and each $C_...

as \left appears at the end of the formula. If you scroll past that you get errors that there is no matching \right. \left and \right must be in matching pairs in the same expression.

If you fix the mismatched \left then you get

! Missing } inserted.
<inserted text> 
                }
l.30  \end{flalign}

as you have $ inside flalign which is wrong as falign already enters math mode.

If you fix the nested math you get the error

! LaTeX Error: There's no line here to end.

as \\ appears but can not force a linebreak where there is no line, so deleting these, which will also remove the warnings

Underfull \hbox (badness 10000) in paragraph at lines 18--22

which are from \\ mis-placed at the ends of paragraphs

You then get the error

! Missing $ inserted.
<inserted text> 
                $
l.44  Fact: $C(n,r)$ = \Bigg(\overset{n}{r}

as you finished the math before the = so the math mode commands following the = generate errors.

Fixing that finally produces an error free log. It is only at this point you should look at the generated PDF as TeX makes no attempt to make sensible PDF output after an error, it just recovers enough to syntax check the remaining document it does not make usable output if you scroll past an error.

Changing $$ to \[ then results in this document

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
%\documentclass[fleqn]{article}
%\setlength{\mathindent}{0pt}
\begin{document}


\section{1 Limit of a Riemann Sum}

Suppose $f$ is defined on [$a,b$] and $L$ is a real number. We write
\[\int_{a}^{b} f(x) dx = \lim_{|| P ||\to\ 0} \sum\limits_{k=1}^n f(C_k)\Delta \chi _k = L\]
if for every $\epsilon$ $>$ 0, there is a  $\delta$ $>$ 0 so that if $P$ is any partition 
of [$a,b$] with $\lVert P \rVert < \delta$ and each $C_k$ is a number
in the $k$th subinterval of $P$, then

\[\left\vert \sum\limits_{k=1}^n f(C_k)\Delta \chi _k - L \right\vert < \epsilon\]

Note: The "variable of integration" is irrelevant.
\[\int_{-2}^{5} x^3 dx = \int_{-2}^{5} y^3 dy = \int_{-2}^{5} t^3 dt = ...\]

 Ex: Do the following exist?
 \begin{flalign}
 \int_{1}^{3} \frac{1}{x^2} dx &&
  \\&&
 \int_{0}^{2} \frac{1}{x^2} dx&&
 \end{flalign}
 Ex: Evaluate $$\int_{1}^{4} f(x) dx$$ where $f(x)$ = 
 \[\begin{cases}
 3x^2 + 1, \quad 1 \leq x \leq 3
 \\
 28, \quad \quad \quad \quad 3<x \leq 4
 \end{cases}
 \]

 You may use the fact that $$\int_{1}^{3} x^2 dx = \frac{26}{3}$$ where $f(x)$ 


 \section{Miscellaneous Trivia}
 Prove: If A
\[\subseteq B, then \quad A \cap C \subseteq B \cap C\]

 Fact:
\[C(n,r) = \Bigg(\overset{n}{r}\Bigg) = 
\frac{P(n,r)}{r!} = 
\frac{\frac{n!}{(n-r)!}}{r!} = 
\frac{n!}{r!(n-r)!}\]
\end{document}

which is

adding fleqn to the \documentclass will make the displayed expressions flush left and also all expressions such as $\epsilon$ $>$ 0 should be typeset as a single math expression so you get the correct spacing and the final 0 is inside the math so $\epsilon > 0$, Also never have a blank line before a display math.

Finally get this which may be an approximation to your required layout, I had to make some guesses

\documentclass[fleqn]{article}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
%\documentclass[fleqn]{article}
%\setlength{\mathindent}{0pt}
\begin{document}


\section{1 Limit of a Riemann Sum}

Suppose $f$ is defined on [$a,b$] and $L$ is a real number. We write
\[\int_{a}^{b} f(x) dx = \lim_{|| P ||\to\ 0} \sum\limits_{k=1}^n f(C_k)\Delta \chi _k = L\]
if for every $\epsilon > 0$, there is a  $\delta >0$ so that if $P$ is any partition 
of $[a,b]$ with $\lVert P \rVert < \delta$ and each $C_k$ is a number
in the $k$th subinterval of $P$, then
\[\left\vert \sum\limits_{k=1}^n f(C_k)\Delta \chi _k - L \right\vert < \epsilon\]

Note: The "variable of integration" is irrelevant.
\[\int_{-2}^{5} x^3 dx = \int_{-2}^{5} y^3 dy = \int_{-2}^{5} t^3 dt = ...\]

 Ex: Do the following exist?
 \begin{gather}
 \int_{1}^{3} \frac{1}{x^2} dx\\
 \int_{0}^{2} \frac{1}{x^2} dx
 \end{gather}
 Ex: Evaluate \[\int_{1}^{4} f(x) dx\] where
\[f(x) = 
 \begin{cases}
 3x^2 + 1, \quad 1 \leq x \leq 3\\
 28, \quad \quad \quad \quad 3<x \leq 4
 \end{cases}
 \]

 You may use the fact that
\[\int_{1}^{3} x^2 dx = \frac{26}{3}\]
 where $f(x)$ 


 \section{Miscellaneous Trivia}
 Prove: If 
\[A\subseteq B, then \quad A \cap C \subseteq B \cap C\]

 Fact:
\[C(n,r) = \Bigg(\overset{n}{r}\Bigg) = 
\frac{P(n,r)}{r!} = 
\frac{\frac{n!}{(n-r)!}}{r!} = 
\frac{n!}{r!(n-r)!}\]
\end{document}