Integration Definition and 1000 Threads

Can you please help me ? I have tried to do it many times but still got the wrong answer. Thank you in advance.

Hi, I have this formula ## f(\theta, \phi) = \frac{sin \theta}{4\pi}## I have this statement that say if I integrate this formula above on a sphere then p = 1. what does integrate on a sphere means? I know ##\int_0^{2\pi} ## is used for the circle.

The detailed list of the concepts I should master I'm attending the last year of high school and I'm currently studying limits. For university test reasons I'll need to study on my own topics such as differentiation and integration... and I have just 14 days to do so! Firstly, do you think it's...

Performing the x-integration first the limit are x=y2 and x= -y2 and then the y limits are 0 to 1. This gives the final answer 2/5 But i am getting confused when trying to reverse the order of integration. My attempt is that i have to divide the region in 2 equal halfs and then double my answer...

Hi, I'm trying to integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}## and ## \int \frac{pdz \cdot \hat p}{(\sqrt{p^2 + z¸^2)^3}}## I get ## \frac{\hat z}{(\sqrt{p^2 + z¸^2)}}## and ## \frac{2z \cdot \hat p}{(\sqrt{p^2 + z¸^2)}}## But the correct answer should be ## \frac{z \hat...

Hi, I'm wondering how can I get ## \phi(t) = A sin(\omega t) + B cos(\omega t)## I know I have to integrate 2 times ##\ddot\phi = -\omega^2\phi##. However, I don't have any more explanation in my book. I know A and B are the constants of integration.

$$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$ So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise. The generalized complex integral would be $$\int_{C} \frac{e^{-i \alpha...

Need some help on how to solve the integration formula for Maxwell speed distribution, here is the procedure on how to solve for the kinetic energy: Not familiar with the error function yet, but the result for the kinetic energy integration is...

I am looking for a (practice)book that has problems on definite and indefinite integration from easy to intermediate. also which book covers the prerequisites of calculus for books like Griffiths.(similar to the topics in chap 1 of Griffiths but more in-depth)

Good day I have a problem figuring out the surface of integration according to the exercice, we have a paraboloid that cross a disk on the xz plane, the parabloid cross the xz plane on a smaller disk r=√3/3 so for me after going to the final step of integration and using polar coordinate i...

my Problem is that I get a different result when I switch the order of integration (X over Y), I couldn't spot the mistake, any help would highlyu appreciated

Find all continuous functions $f:[1,\,8] \rightarrow \mathbb{R} $ such that $\displaystyle \int_1^2 f^2(t^3)dt + 2\int_1^2 f(t^3)dt=\dfrac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt$

i would appreciate alternative method...

The tables and Wolfram Alpha say ## \int dx/(x^2 + a^2) ^{1/2} = log~ [( x^2 + a^2)^{1/2} + x] ##. So if a=0 we get as answer ## log (x + x) = log( 2x) ## if ## (x^2)^{1/2} = x, or ## log (x - x) = log( 0) ## if ## (x^2)^{1/2} = -x but surely ## \int dx/(x^2)^0.5 = \int...

by observation I choose (c) since the limit values may not be =

Hello, I want to convert a summation in reciprocal space and I am unsure about the integration volume. I have started with the formula: $$\sum_{\vec{k}} \rightarrow \frac{V_{k}}{(2\pi)^{3}}\int\int\int \mathrm{d}V_{k}$$ where: $$\mathrm{d}V_{k} = k^{2}\mathrm{d}k...

From the equations, I can find Jacobians: $$J = \frac {1}{4(x^2 + y^2)} $$ But, I confuse with the limit of integration. How can I change it to u,v variables? Thanks...

I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...

I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...

Summary:: I want to iterate a mathematical model using a programming language. The equation of the mathematical model is simple. The following is a brief explanation. I want to iterate a mathematical model using a programming language. The equation of the mathematical model is simple. The...

Converting to a polar integral : Integrate ##\(f(x, y)=\) \(\left[\ln \left(x^{2}+y^{2}\right)\right] / \sqrt{x^{2}+y^{2}}\)## over the region ##\(1 \leq x^{2}+y^{2} \leq e\)## So, \begin{array}{c} 1 \leq x^{2}+y^{2} \leq e \\ 1 \leq x^{2} \leq e \quad 1 \leq y^{2} \leq e \\ 1 \leq x \leq...

write on Interpretations of integration (give examples).

what is The meaning of integration?

The area of two lines that I need to find is 2.36, however i need this in exact form. The lines are y=-x/2e+1/e+e the other line is y=e^x/2 Since y=-x/2e+1/e+e is on top it is the first function. A=(the lower boundary is 0 and the top is 2) -x/2e+1/e+e-e^x/2 If you could please help!

For a double integral, we might treat the "inner integral" separately and be able to compute something like ##\int_{x_1}^{x_2} f(x,y) dx## by holding ##y## constant during the integration. The same technique is applied in other places too, like for solving exact differential equations. I haven't...

can integrate 1/(x*lnx) by parts??

Hey Could you give me a hint how to explain this example? Need help to prove statement in red frame. Example from book (Topics In Banach Space Integration) by Ye Guoju‏، Schwabik StefanThank you

If i want to calculate the volume of a cone i can integrate infinitesimal disks on the height h of the cone. I was told that if i want to calculate the surface of the cone, this approximation is not correct and i have to take the slanting into account, this means that instead of...

Let x=t^2 Then dx=2t dt Integral of 1/(x(1-x))^(1/2)dx = integral of 2tdt/t(1-t^2) ^(1/2) = integral of 2dt/(1-t^2) ^(1/2) = 2 arcsin(t) +c = 2 arcsin(rt(x)) +c. But the answer in my book is arcsin(2x-1) +c. Tell me how 2 arcsin(rt(x) +C= arcsin(2x-1) +c I know the constant will vary for both...

The Gauss-Kronrod quadrature uses the zeros of the Legendre Polynomials of degree n and the zeros of the Stieltjes polynomials of degree n+1. These zeros are the nodes for the quadrature. For example using the Gauss polynomial of degree 7, you will need the Stieltjes of degree 8 and both makes...

In Thermodynamics, I have seen that some equations are expressed in terms of inexact differentials, ##\delta##, instead of ##d##. I understand that this concept is introduced to point out that these differential forms are path-dependent, although I am not clear how they can be handled. So, are...

##\int \frac{e^x (2-x^2)}{(1-x) \sqrt{1-x^2}} dx## I tried using substitution x = sin θ but still can not solve it. I guess I have to get rid the term ex but do not know how Thanks

The integral of cothx is ln|sinhx|+C. Does this mean the integral of coth2x is ln|sinh2x|+C? If not, does anyone have a link to a page on how it is achieved - I'm trying to compile a list of all common hyperbolic function derivatives and integrals. However, I can't find anything to confirm if...

In spherical poler coordinates the volume integral over a sphere of radius R of $$\int^R_0\vec \nabla•\frac{\hat r}{r^2}dv=\int_{surface}\frac{\hat r}{r^2}•\vec ds$$ $$=4\pi=4\pi\int_{-\inf}^{inf}\delta(r)dr$$ How can it be extended to get $$\vec \nabla•\frac{\hat r}{r^2}=4\pi\delta^3(r)??$$

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...

I can only find a solution to \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel...

Summary:: I just need to know how we got from the 'beginning point' to the 'end point'/'answer'. The left side is where we start and my professor did a bunch of calculations so fast that I wasn't able to understand how he got the result on the right side. Could someone help me integrate this...

I can only find a solution to \int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . The closed form solution to \int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho I am not able to find anywhere. Is there any way in which I can approach this problem from scratch...

I know that the formula for volume is equal to the definite integral ∫A(x)dx, where A(x) is the cross sectional. I found the definite integral where b=5 and a=0, for ∫4x2dx. I obtained the answer 500/3, however this was incorrect, and I'm unsure of where I went wrong? Thank you.

The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.

EQ 1: Ψ(x,0)= Ae-x2/a2 A. Find Ψ(x,0) So I normalized Ψ(x,0) by squaring the function, set it equal to 1 and getting an A I. A=(2/π)¼ (1/√a) B. To find Ψ(x,t) EQ:2 Ψ(x,t)= 1/(√2π) ∫ ∅(k) ei(kx-ωt)dk --------->when ω=(ħk2)/2m and integral from -∞ to +∞ EQ 3: ∅(k)= 1/(√2π) ∫ Ψ(x,0)...

ok I posted a image to avoid any typos but was wondering why the question has dx and options are in dt I picked C from observation but again that was assuming f was a horizontal line of which it could be something else that way the limits stay the same but the area is cut in halfopinions...

Not sure how do I start from here, but do I let $$u = lnx$$ and substitute? Cheers

We have so many great books available for Calculus, such as : Spivak's Calculus, Stewart Calculus, Thomas Calculus , Gilbert Strang's Calculus, Apostol's Calculus etc. These books are very nice but they teach you the concepts well and all the standard techniques that are available for solving...

Homework Statement: The question is in Attempt at a solution. Homework Equations: x=tanA/b I tried by substituting x=tanA/b but it did'nt helped.Now I cannot think of any other thing to do.Help.

I was trying to solve a differential equation that I defined to study the dynamics of a system. Meanwhile, I encounter integration. The integration is shown in the image below. I tried some solutions but I am failed to get a solution. In one solution, I took "x" common from the denominator terms...

For the diagram In scalar field theory, I have obtained an integral which looks like $$\int_{0}^{\Lambda} \frac{d^4 q}{(2\pi)^4} \frac{i}{q^2 - m^2 + i\varepsilon} \frac{i}{(p - q)^2 - m^2 + i\varepsilon}$$ I am required to calculate this and obtain the divergent amplitude $$i\mathcal{M} =...

There has been a demonstration of SiC BJT ADCs, and SiC JFET SRAM. What would be the major limitations associated with higher forms of integration for SiC? For example, a 1billion transistor CPU based on SiC?