2
votes
2answers
46 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
0
votes
1answer
42 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
0
votes
1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
1
vote
1answer
12 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
5
votes
1answer
84 views

Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $

there is integral $$ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}} $$ but have no idea ...
1
vote
0answers
14 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
1
vote
1answer
32 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
3
votes
3answers
72 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
2
votes
1answer
27 views

Using Polar Integrals to find Volume of surface

Here's the Question and the work that I've done so far to solve it: Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z ...
4
votes
0answers
56 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$
1
vote
2answers
132 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
0
votes
2answers
46 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
0
votes
3answers
30 views

How to get from $3\int_{-1}^0 (x^3-x) dx \,\,\,- \,\,\, 3\int_0^1 (x^3-x) dx$ to $6\int_{-1}^0(x^3-x)dx$?

Homework problem: Set up the definite integral that gives the area of the region. Two functions are given: $y_1 = 3(x^3-x)$ $y2 = 0$ The graph of $y1$ runs from x=-1 to x=1. I've gotten this ...
2
votes
1answer
87 views

Simple integral $\displaystyle\int \frac{e^x}{x^2-a^2}\ dx$

Is this integral solvable? $$\int \frac{e^x}{x^2-a^2}dx,\quad a>0.$$
1
vote
4answers
70 views

How can I prove the integral?

Prove that $$ \int\frac{dx}{x(\log_e x)^{7/8}} = 8(\log_e x)^{1/8} $$ I am totally lost on this subject. Any help how to prove this is appreciated!
1
vote
1answer
23 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
0
votes
2answers
32 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
0answers
15 views

Find the volume of this region.

Find the volume of the region bounded by $ (x^2+y^2+z^2)^2=x$. I do not know how to deal with this question. Please help.
3
votes
2answers
136 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
0
votes
1answer
44 views

Finding the area between 2 curves using Green's Theorem

Find the area bounded by $y=x^2$ and $y=x$ using Green's Theorem. I know that I have to use the relationship $\int_c Pdx + Qdy = \int\int_D1dA$. But I don't know what my boundaries for the integral ...
2
votes
2answers
67 views

Prove that integral of continuous function is continuously differentiable

Lots of things going on here. I immediately know that $F(x)$ does exist since $f$ is riemann integrable due to the fact that it is continuous. First I need to show that $F$ is continuous, then find ...
1
vote
1answer
18 views

How do I calculate the area under a curve using the midpoints of rectangles?

I figured out how to calculate the area under the curve from the Right endpoint and Left endpoints, but I can't figure out how to calculate it using the midpoints. Especially when it says $M_3$. Ill ...
2
votes
4answers
49 views

How to calculate this area in $\mathbb{R}^2$?

Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$ First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what ...
0
votes
0answers
37 views

Line Integral and Green's Theorem

I have been working on a simple line integral: line integral of $x\,dy+y\,dx$ (I don't know how do write this properly, I'm sorry!) over the closed curve enclosed by the the ellipse $x^2+5y^2=4$ and ...
0
votes
2answers
30 views

Double integral with integration by parts

This is for an online web assignment with multiple choice. $ \int_1^2 \int_0^1 xye^{(x^{2}+1)y} $ I solved the inner integral with respect to y: $\int_0^1 xye^{(x^{2}+1)y} $ = ...
6
votes
4answers
265 views

Does the improper integral exist?

I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...
1
vote
1answer
43 views

positive measurable function on $[0,1]$

If $f$ is a positive measurable function on $[0,1]$, which is larger, $$\int_{0}^{1}f(x)\,\log f(x)\,dx \qquad \text{or} \qquad \int_{0}^{1}f(s)\,ds\int_{0}^{1}\log f(t)\,dt$$ Can you help me ...
0
votes
1answer
16 views

Linear algebra with integrations and primitive functions. What is the area?

The line $y = kx + m$ tangents the line $y= \frac12x^3$ where $x = -2$. Decide the area of the area that is limited by the lines and the curve with the help of integration. (or any other way with ...
0
votes
0answers
16 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
1
vote
4answers
57 views

Integrating a function with substitution

Totally forgot how to integrate. $$ \int \frac{1}{x^2 \sqrt{x^2+4}}dx$$ Just need a tip, for this what would I use to substitute?
1
vote
1answer
24 views

Line integral segment of parabola

Suppose $$ \vec{F} = \nabla f(x,y) = 6y \sin (xy) \vec{i} + 6x \sin (xy) \vec{j}, $$ and C is the segment of the parabola $y = 5 x^2$ from the point $(2,20)$ to $(6,180)$. Then, what is $$\int_C ...
1
vote
2answers
111 views

integral of sin(x) to the power 2014

For a course in Complex Analysis we're tasked to find the integral of \begin{align*} \int_0^{2 \pi} (\sin\theta)^{2014} d \theta \end{align*} but I'm a bit stumped so far on how to do this. What I've ...
0
votes
0answers
23 views

Surface Area cut from cone by cylinder

I genuinely have no idea how to do this. I've been struggling on this problem for about two hours now (more hours last night), and am not getting anywhere. The question is: Find the surface area ...
2
votes
2answers
62 views

Simple algebra in a differential equation.

I have the differential equation: $$\frac{dy}{dx}=\sin (x-y).$$ Substituting $v=x-y$ and $dy=dx-dv$, I got down to the equation:$$\frac{dv}{1-\sin(v)}=dx.$$ Multiplying the LHS by $\frac{1+\sin ...
0
votes
3answers
47 views

Anti-derivative of constant to a power

How do I approach a problem like this? $\int xe^{x^2 - 1} dx$? I tried U-substitution, but what confuses me is that $x$ is not the first derivative of $x^2 - 1$ $\frac{du}{dx} x^2 - 1 = 2x$ $du = ...
0
votes
2answers
44 views

Evaluating improper integral

Im trying to evaluate the improper integral $$\int_{0}^{\infty}\left( \frac{e^{i \omega t}+e^{-i \omega t}}{2}\right) e^{-st} dt$$, where $\omega$ and $s$ are real positive constants and ...
0
votes
1answer
18 views

Help on a homework question finding the area of a curve using polar equations

Show the area enclosed by $r=a(p+qcos\theta)=\dfrac{2p^2+q^2}{2}\pi a^2$. Due to the form of the equation, the curve is either a cardioid or an egg shape - either way, the boundaries of the integral ...
1
vote
1answer
58 views

Integration w/ Change of Variables

folks. I've got this question: Let $D$ be the region $\{(x,y) ~|~ 0 \leq y \leq x, 0 \leq x \leq 1\}$. Evaluate: $$\iint_D (x + y) dxdy$$ by making the change of variables $x = u + v$, $y = u ...
1
vote
1answer
84 views

Integrate $\int_0^1 \frac{\ln x}{\sqrt{1-x^2}}dx$. HW

Since this is a homework problem, a hint would be appreciated to help me get this started, since I have no idea how to start this. Thanks Here's the problem: Compute the improper integral: $$\int_0^1 ...
1
vote
2answers
38 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
1
vote
1answer
30 views

Weird question about probability density function

I'm assuming "actual" means the total probability of the PDF (the integral from $-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ...
1
vote
1answer
37 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
1
vote
2answers
39 views

Help with separable differential equation? $\frac{dy}{dx} =2y^2$

I'm new to separable differential equations, and I'm stuck on this question: $\frac{dy}{dx} =2y^2$ Using the initial condition $y(2)=3$, find $y(1)$. So far I've integrated to get $\frac{dy}{dx} ...
1
vote
3answers
48 views

Lebesgue measure problem

Let $f$ be a non-negative measurable function on $\mathbb{R}$, and suppose that $\int f=0$. Prove that the set where $f \neq 0$ is a zero set. The hint says to let $E_n=\{f>1/n\}$ and then compare ...
0
votes
1answer
38 views

Double Integrals - Volume of the region

Use double integrals to calculate the volume of the following region. The solid beneath the cylinder $z=y^2$ and above the region $R = \{ (x,y) : 0 \leq y \leq 1, y \leq x \leq 1 \}$. I just need ...
4
votes
1answer
47 views

Show that $g\in\mathcal{L}^q(\mu)$.

Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that $$\int |fg|d\mu\leq C\|f\|_p$$ for ...
1
vote
1answer
45 views

Finding a Differential Equation that Satisfies an Initial Condition

Find the solution of the differential equation that satisfies the given initial condition: $$ \frac{dL}{dt} = kL^2ln(t), L(1) = -8$$ The thing that's really screwing me up here is that darn k. I've ...
0
votes
0answers
35 views

Stuck with trig substitution

I am stuck with problem at my homework assignment. $$\int \sqrt{1+4x^2}dx$$ I try to apply trigonometric substitution $$x = \frac 1 2\tan{2u}$$ $$dx = \frac 1 {\cos^2{2u}}du$$ But after ...
1
vote
1answer
78 views

Gaussian Quadrature -Deriving a Formula-

eThe following is an exercise in the problem section of the Gaussian Quadrature chapter. The theorem: Derive a formula of the form $$\int_{a}^{b} f(x)dx \approx w_0f(x_0) + w_1f(x_1) + w_2f'(x_2) ...
1
vote
0answers
19 views

Continuity of a function on a box $Q$ implies integrability

Let $Q \subseteq \mathbb{R}^n$ be a box, and say $f: Q \to \mathbb{R}$ is continuous, then $f$ is integrable on $Q$. MY ATTEMPT: Since $f$ is continuous on $Q$, then $f$ must be uniformly continuous ...