As we know, the integral of function $f:[a,b]\to \mathbb{C}$ is defined $$ \int\limits_a^bf(t)\,\mathrm{d}t=\int\limits_a^b\operatorname{Re}[f(t)]\,\mathrm{d}t\,+\,i \int\limits_a^b\operatorname{Im}[f(t)]\,\mathrm{d}t$$ But, I am curious to know how we defined this integral in this way? Why? As integral of any function $g:[a,b]\to \mathbb{R}$ measures area. But I also want to know what the integral of $f$  measures?
27772
In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$.  But what is being measured with complex line integration, when you go from a point $z_1$ to a point $z_2$ in the complex plane?  With regular line integration I can see $f(x,y)$ maps $(x,y)$ to a point on the $z$ axis directly above above/below $(x,y)$.  But in the complex case, when you map from the domain $Z$ to the image $W$, you are mapping from $\mathbb{R^2}$ to $\mathbb{R^2}$ ...it is not mapping a point to 'directly above/below'...so I don't have any intuition of what is happening with complex line integration.
43637
Say I have a function $f(x)$ on some interval $[a,b]$.  Say it is integrable such that $\displaystyle\int f(x)~dx $ is defined.  Is $\displaystyle\int f(x)~dx $ necessarily continuous?  If I were to know that the integral is integrable itself such that $\displaystyle\int \int f(x)~dx $ is defined. Would that change anything?   If so why?  Thank you
111937
I'm doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can't come to the conclusion. Can anyone help me. I really appreciate.          Consider a measure space $(X, M, \mu)$.Suppose $\{f_{n}\} \in L^{+}, f_{n} \rightarrow f$ pointwise and $\int f = \lim \int f_{n} &lt; \infty$. Then $\int_{E} f = \lim \int_{E} f_{n}$ for all $E \in M$. However, this need not be true if $\int f = \lim \int f_{n} = \infty$      Thanks so much.
117979
From my understanding if the integrand is 1, then it gives you the volume of the region defined by the bounds. But what does the value of a triple integral represent if the integrand is a function for a surface in space?
46758
Let us define a very simple integral:   $f(x) = \int_{a}^{b}{x}$    for $a,b\ge 0$.  Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$?  I drew the graphs and thought about it but to me integration, at least in two-dimensions, is just taking the area underneath a curve so why does it matter which direction you take the sum?
90648
What are all the translation invariant measures on $\mathbb{R}$?  Except Lebesgue measure on $\mathbb R$ I didn't find any translation invariant measure. So I put this question?  I know that if $\mu$ is a measure then $c \times \mu$ is again a measure where $c&gt;0$.
117243
Find out the value of $a, b$ and $c$, for which $$ \lim _{x\to 0}\:\:\frac{1}{x^5}\left(\int _0^x\:\left(a+b \cos\left(t\right)+c \cos\left(2t\right)\right)dt\right)=\frac{1}{5} $$  My first thought was to separate the integral into multiple ones because i thought that it is pretty clear that each integral is a common one and a,b,c are constants and can be taken out . After I solved the integral, that's where I had the problem. I couldn't do the limit of the result that I've got in order to find out the value of $a,b$ and $c$ .
69661
Let $f(x)$ be continuous on $[a,b]$ and suppose $\int_a^b f(x)g(x)dx = 0$ for every continuous function $g$ on $[a,b]$. Prove that $f(x)=0$ on $[a,b]$.   I understand that $f(x)$ must be zero otherwise the integral can't possibly be zero for all possible $g$. But, I am not sure how to approach this systematically. Any suggestions would be appreciated. Thanks!
67558
What is the historical background for this term?   I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve.  It seems to me a strange choice of word.
39612
How to show $f=g$ almost everywhere  Let two real-valued measurable functions $f$ and $g$ be such that for any measurable set $E$ the integrals $\int_E f\,d\mu$  and  $\int_E g\,d\mu$ coincide.  Could you please help.  It seems obvious. we could define the integrals as charges. can we do sth from here. but I do not know how to approach.
18829
I have a question:     Suppose $f$ is continuous and even on $[-a,a]$, $a&gt;0$ then prove that   $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$   How can I do this? Don't know how to start.
105003
If $f \colon [a,b]\rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$.  I think that I have to proceed with the Mean Value Theorem of Integrals...
43906
I have a question that I need help with getting started (possibly I would be back for more help).     I have a measure space $(X,A,\mu)$ that is finite, and $f \in L^{\infty}(\mu)$. Also, defined is $a_n = \int_X\,|f|^n\,d\mu$. I need to show that the limit is: $$\lim_{n\to\infty}\,\frac{a_{n+1}}{a_n} = \|f\|_{\infty} .$$   I am stuck on getting started, anybody have any suggestions?  thanks much
77431
If $f$ is bounded by $M$ on $[a, b]$ and if the restriction of $f$ to every interval $[c, b]$ where $c$ in $(a, b)$ is Riemann integrable, then $f$ is Riemann integrable and that $\int _c^b f \to \int_a^b f$ as $c \to a+$.  I am stuck with this problem.
120818
Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}&gt;0: \Bigg|x^p\frac{d^q}{d x^q}f(x)\Bigg|&lt; C_{pq}$$ I was wondering whether, if $\forall p\in\mathbb{N_{\ge 0}}\quad\int_{\mathbb{R}}x^pf(x)d\mu=0$, then $f$ is constantly, or almost everywhere, null. I cannot find a counterexample and therefore I think that the implication might well hold, but I cannot prove it either.  Since $x^p f(x)$ belongs to $S_\infty\subset L_1$, thanks to the fact that $f$ belongs to it, and is continuous I think that the Lebesgue integral and the Riemann improper integral $\mathscr{R}\int_{-\infty}^\infty t^pf(t)dt$ are the same. Please correct if I am wrong.  Nevertheless I cannot use calculus facts to prove the desired implication... I have also tried using the fact that the Fourier transform induces a bijection $S_\infty\to S_\infty$, but with no result. What do you think about it? Has anybody got a counterexample or proof? Thank you very much!
107150
Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function. Show that     $$ \int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)&gt;t\right\})\, dt. $$     (The right integral is to be read as an Lebesgue-integral.)      Hello, my idea is to approximate $f$ by the functions $$ f_n:=2^{-n}\sum\limits_{k=1}^{\infty}\chi_{\left\{x\in X: f(x)&gt; k/2^n\right\}}. $$  With this approximation by simple functions I got $$ \int\limits_X f\, d\mu=\lim\limits_{n\to\infty}\int\limits_X f_n\, d\mu=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{\infty}\frac{k}{2^n}\cdot\mu\left(\left\{x\in X: f_n(x)=k/2^n\right\}\right)\\=\lim\limits_{n\to\infty}\sum\limits_{k=1}^{\infty}\frac{k}{2^n}\cdot\mu\left(\left\{x\in X:f(x)&gt;k/2^n\right\}\right) $$  This is the point where I do not come along anymore... is that right to this point and if yes: How can I continue in order to get the desired right side?  With regards
29793
Assmue that $f(x)\in C(0,+\infty)$ and for any $x&gt;0$ have $$\int_{x}^{x^2}f(t)dt=\int_{1}^{x}f(t)dt$$  Find this $f$  My try: we have $$2xf(x^2)-f(x)=f(x)\Longrightarrow xf(x^2)=f(x),\forall x&gt;0$$ and then I can't works