All questions with tag [math: integration]


Is the Lebesgue integral the completion of integrals on step functions?

Lierre gave a very helpful insight at answer 5 on about what Riemann integrals are. My question relates to whether this can be extended to Lebesgue integrals. Lierre pointed out that Riemman integrals can be seen as the natural extension of the 'obvious' linear form on characteristic (or 'indicatrix') functions on real line intervals. There i...
2022-07-25 17:47:17

Why integrals with respect to different variables aren't equal?

I have a function $y=x^2+1$, the integral from $-1$ to $2$ is $\int_{-1}^{2}(x^2+1)dx = 6$. The function $x=\sqrt{y-1}$ is the same as the above function. The integral would be from $0$ to $(2)^2+1=5$. So I thought that $\int_{0}^{5}(\sqrt{y-1})dy$ would be equal to the first one. But it turns out that it does not. The integral of the second fun...
2022-07-25 17:47:13

Surface integral (stokes?)

I intend to address the adhering to trouble, I intend to locate $$ \iint_S x \, \mathrm{d}S $$ where S is the component of the allegorical cyndrical tube that exists within the cyndrical tube $z = x^2/2$, and also in the first octant of the cyndrical tube $x^2 + y^2 = 1$ I was clearly thinking of switching over to round works with, yet I have ...
2022-07-25 17:47:02

Simple proof of integration in polar coordinates?

In every instance I saw of integration in polar coordinates the Jacobian component is made use of, not that I have a trouble with the Jacobian, yet I asked yourself if there is a less complex means to show this which will certainly additionally offer me some even more instinct concerning the Jacobian. If I attempt to merely write the differenti...
2022-07-25 17:47:02

Weird integral with cylinders

I have this unusual indispensable to locate. I am in fact searching for the quantity that is defined by these 2 formulas. $$x^2+y^2=4$$ and also $$x^2+z^2=4$$ for $$x\geq0, y\geq0, z\geq0$$ It is an unusual object that has the aircraft $z=y$ as a divider panel for both cyndrical tubes. My troubles is that I can not locate the integration lim...
2022-07-25 17:46:47

Why does this integral rearrangement hold?

A remedy to among my technique troubles entails this: $\int^\infty_0 \{\int^x_0 dy\}f(x)dx = \int^\infty_0 \{\int^\infty_y f(x) dx\}dy $ Where f() is a PDF function of a continual arbitrary variable (if that makes any kind of distinction). Why does this job? And also what should I recognize in order to have the ability to use this sort of a...
2022-07-25 17:46:18

Estimate the area restricted by $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$.

I need to estimate the area between the functions $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$. where $a>1$. Now I have tried quite a few ways to do this, but nothing comes to mind. I tried writing out the taylor series, I tried changing this around. Nothing really gave a decent approximation. A decent in my mind would be anyth...
2022-07-25 17:44:17

Evaluate definite integral $\int_{-1}^1 \exp(1/(x^2-1)) \, dx$

How to review the adhering to precise integral: $$\int_{-1}^1 \exp\left(\frac1{x^2-1}\right) \, dx$$ It appears that uncertain indispensable additionally can not be shared in typical features. I would certainly such as any kind of remedy in preferred primary or non - primary features.
2022-07-25 17:41:51

How to find out what changes applied to integral?

I have actually obtained such indispensable $$\int{\frac{\sqrt{x^2+1}}{x+2}dx}$$ and also with Maple I obtained something similar to this: $$\int\frac{1}{2} + \frac{1+3u^2+4u^3}{-2u^2+2u^4-8u^3}du$$ And I need to know just how to achive this adjustments. I attempted to make use of WolframAlpha, yet there is scarier remedy. This indispensable wa...
2022-07-25 17:41:22

How to calculate the integral in normal distribution?

The manufacturing facility is making items with this normal distribution: $\mathcal{N}(0, 25)$. What should be the maximum mistake approved with the chance of 0.90? [Result is 8.225 millimetre] How will I compute it? Just how to incorporate: $\exp\left(- \frac{x^2}{2} \right)$?
2022-07-25 17:38:52

Integral of $x^2\ln(x)$ using Simpson's rule

This is my homework question: Calculate $\int_{0}^{1}x^2\ln(x) dx$ using Simpson's formula. Maximum error should be $1/2\times10^{-4}$ For solving the problem, I need to calculate fourth derivative of $x^2\ln(x)$. It is $-2/x^2$ and it's maximum value will be $\infty$ between $(0,1)$ and I can't calculate $h$ in the following error formula for u...
2022-07-25 17:21:06

An integral involving trigonometric functions and its inverse

I have tried to evaluate the following integral for the last few hours, and I did not succeed: $$ \int\limits_{0}^{2 \pi} e^{\mathrm{i} \cdot n \cdot\mathrm{arcsin}(r \cdot\mathrm{sin}(\theta))} \cdot e^{\mathrm{i}\cdot m \cdot \mathrm{arcsin}(r \cdot \mathrm{cos}(\theta))} d \; \theta$$ for $0<r<1$. And also this other integral: $...
2022-07-25 17:19:19

Change of variables formula

Consider an example. Let $f(x)$ be a function on the unit sphere $S^{n-1}$. Consider an integral $$ \int\limits_{S^{n-1}} f(x) \, dx $$ I want to make a substitution $x = x_{0}t + \sqrt{1-t^2}y$, where $t \in [-1,1]$, $y \bot x_{0}$, $|y| = 1$. After change of variables the manifold of integration changes. In general case if I have an integral...
2022-07-25 17:17:03

What is the $x(t)$ function of $\dot{v} = a v² + bv + c$ to obtain $x(t)$

How to solve $$\frac{dv}{dt} = av^2 + bv + c$$ to obtain $x(t)$, where $a$, $b$ and $c$ are constants, $v$ is velocity, $t$ is time and $x$ is position. Boundaries for the first integral are $v_0$, $v_t$ and $0$, $t$ and boundaries for the second integral are $0$, $x_{max}$ and $0$, $t$.
2022-07-25 17:16:52

contour integral with rational and cosh

Here is a fun looking integral. $$\int_{0}^{\infty}\frac{1}{(4x^{2}+{\pi}^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}$$. I rewrote it as $\frac{2e^{z}}{(4z^{2}+{pi}^{2})(e^{2z}+1)}$ It would appear there is a pole of order 2 at $\frac{\pi i}{2}$. This is due to it being a zero of cosh and the rational part. I think the residue at $\frac{\pi i}{2}$ is ...
2022-07-25 17:14:36

Finding $f'(x)$ when $f(x)=\int^1_0 e^{xy+y^2}dy$

If $f(x) = \int^1_0 e^{xy+y^2}dy$, find $f'(0)$. I understand that this is function defined by an integral, and $e^{y^{2}}$ does not integrate into an elementary function. So, I will need to take $f'(x)$ which yields: $$\int^1_0 ye^{xy+y^2}dy$$ I am trying to integrate this, but I am failing. I take it I should use integration by parts, but I ...
2022-07-25 17:14:28

Probability that a Multivariate Normal RV lies within a Spherical Region of Radius R

I am currently using different procedures to estimate the probability that a $D$-dimensional Gaussian random variable with mean $\mu$ and covariance $\Sigma$ lies within a sphere of radius $R$ that centered about the origin. That is, I am estimating $P(|| X ||_2 < R)$ where $X \sim N(\mu, \Sigma)$ and $X \in \mathbb{R}^D$. I am wond...
2022-07-25 17:13:11

del operator - partial derivatives

I'm taking a class in Electromagnetism, and also I'm learning more about the partnerships in between voltage and also an electrical area from Faraday - Maxwell formulas. The formula I have problem with is:. $$E = -\nabla V$$ where $E$ is the electrical area (a vector), $\nabla$ is the slope driver, and also $V$ is the voltage (a scalar). Offer...
2022-07-25 17:08:13

Show velocity of a particle during its flight at time $t$

I'm completely stuck, I think I have to use Newton's second law but I have no idea where to start, any help would be appreciated! At time $t=0$ a particle of unit mass is projected vertically upward with velocity $v_0$ against gravity, and the resistance of the air to the particle's motion is $k$ times its velocity. Show that during its flight t...
2022-07-25 17:05:57

$\int \sin(ax) \,\mathrm d x$

I recognize that $$ \int \sin(x)\,dx=-\cos(x)+C. $$ But I am questioning what will be the $\int \sin(ax)$? I suggest what happens if $x$ is being increased by a constant?
2022-07-25 16:57:46