0
votes
0answers
17 views

How to compute using integration the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around a unit circle?

How to compute the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around the unit circle centered at the origin using the methods of the integral calculus?
0
votes
2answers
17 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
1
vote
4answers
64 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
0
votes
1answer
41 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
0
votes
1answer
35 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
0
votes
5answers
86 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
1
vote
1answer
120 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
1
vote
1answer
26 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
3
votes
1answer
82 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
1
vote
2answers
102 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
4
votes
3answers
115 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
2
votes
1answer
67 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
0
votes
0answers
23 views

Integration of nonlinear and linear ODEs

\begin{equation} \frac{dc_1}{d\tau}= \alpha I(1-c_{0}) + c_{1} (-K_{F} - K_{D}-K_{N} s_{0}-K_{P}(1-q_{0}))+ c_{0}(-K_{N} s_{1}+K_{P}q_{1}), \nonumber \end{equation} \begin{equation} ...
0
votes
1answer
35 views

Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
1
vote
0answers
29 views

Is the “difference-integral” of an integrable function integrable?

assume $f$ is Lebesgue integrable (on $\mathbb{R}$) and $h >0$. Is it true that then the double integral $\int_{-\infty}^{\infty} \int_{x}^{x+h} f(u) \, du \, dx$ always exists? Intuitively I have ...
2
votes
2answers
90 views

How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$?

is there a way to explicitely evaluate this integral for natural numbers $n,m$: $$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$ Apparently, if $n$ is odd, this integral is zero due to symmetry.
1
vote
1answer
45 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
0
votes
3answers
96 views

Assumptions in Word Problems.

My dilemma has been that I am confused on how we make mathematical assumptions in WORD problems. Suppose you are given a related-rates word problem. (Q#) Air is being pumped into a spherical balloon ...
9
votes
1answer
140 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
2
votes
0answers
53 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
2
votes
2answers
56 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
0
votes
0answers
17 views

Finiteness of the lower integral implies finiteness a.e. of the function

I want to prove that if a function $f$ is $\mu$-measurable, $f\geq 0 $ $\mu$-a.e., then the integral of $f$ exists, that is its upper and lower integrals coincide. I've found the proof in Modern and ...
1
vote
0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
4
votes
1answer
174 views

Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
3
votes
1answer
23 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
59 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
3
votes
2answers
243 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
62 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
2answers
41 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
1
vote
1answer
43 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
0
votes
0answers
26 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
3
votes
2answers
80 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
2
votes
1answer
46 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
1
vote
1answer
25 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
1
vote
0answers
20 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
3
votes
2answers
133 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
1
vote
1answer
32 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
2
votes
1answer
46 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
2
votes
4answers
137 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
11
votes
3answers
227 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
5
votes
3answers
120 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
0
votes
0answers
40 views

Definite integral with doable improper case

Is there a way to evaluate one or both of the following integrals: $$ \int_{a_1}^{a_2} e^{ib_1(x+b_2 \sqrt{1+x^2})}dx \quad \text{and}\quad \int_{a_1}^{a_2}\frac{x}{\sqrt{1+x^2}} e^{ib_1(x+b_2 ...
0
votes
1answer
26 views

How to compute the area of this set in the plane?

Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is ...
1
vote
3answers
37 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
4
votes
1answer
122 views

How to integrate these integrals? $\int{\frac{1}{(x^x-x^{-x})}} dx$ [closed]

Problem : $$\int{\frac{1}{(x^x-x^{-x})}} dx$$ I need answer about the following problem. Please help . I will be grateful to you. Thanks.
4
votes
1answer
36 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
2
votes
2answers
164 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
0
votes
1answer
43 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.