Integrals Definition and 1000 Threads

  1. S

    Help with Changing of variables, Jacobian, Double Integrals?

    Homework Statement Show that T(u,v) = (u2 - v2, 2uv) maps to the triangle = {(u,v): 0 ≤ v ≤ u ≤ 4} to the domain D bounded by x=0, y=0, and y2 = 1024 - 64x. Use T to evaluate ∬D sqrt(x2+y2) dxdy Homework Equations The Attempt at a Solution x=u2-v2 y=2uv Jacobian= 4u2+4v2 dudv I guess the...
  2. H

    Problem Involving Maximizing the Ratio of Integrals

    The ratio of integrals ∫〖a(x) b(x)dx〗/ ∫c(x) b(x)dx can be maximized by choosing b(x) equal to the delta function at the point where a(x)/c(x) is a maximum. Can anyone provide the solution for choosing b(x) when b(x) cannot equal the delta function, b(x) is greater than zero with a...
  3. E

    Solving Trigonomic Integrals: Confusion with Prof's Solution

    Homework Statement do my professor did this in class, and it doesn't make sense to me ∫cos^5(x) sin^4(x) dx ∫cos^4(x) sin^4(x) cos(x) dx ∫(1-sin^2(x))^2 sin^4(x) cos(x) dx ∫(sin^4(x) -2sin^6(x) +sin^8(x))cos(x) dx so the above I get but when my professor integrated I became lost, this...
  4. K

    Calculating Iterated Integrals - 2e4

    1. The problem statement, all variables and given/known data Calculate the given iterated integrals ∫02 dy ∫0yy2 * exy dxMy attempt: ∫20dy[exy*y]y0 = ∫20 ey*y*y - ex*0*0 = ∫20ey2*y dx = [ey^2]*y]20 = 2e4 Is this correct?
  5. M

    Interesting integrals, which I think involve the gamma function

    Homework Statement Evaluate the intergrals: a) integral of 3^(-4*z^2) dz from 0 to infinity b) integral of dx/(sqrt(-ln(x))) from 0 to 1 c) integral of x^m * e^(-a*x^n) dx from 0 to infinity Homework Equations gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity The...
  6. M

    W. W. Hansen's Trick to Evaluating Integrals

    In "Probability Theory: The Logic of Science", the author E. T. Jaynes relates that Prof. Hansen at Stanford evaluated integrals by treating constants like pi in an integrand as a variable. Sounds fantastic! Does anyone know how this is done?
  7. K

    Calculating volume using triple integrals

    Homework Statement Find the volume of the solid enclosed between the cylinder x2+y2=9 and planes z=1 and x+z=5Homework Equations V=∫∫∫dz dy dzThe Attempt at a Solution The problem I have here is setting the integration limits. I first tried using: z from 1 to 5-x y from √(9-x2) to -3 x from -3...
  8. A

    What is the topic full of inequalities of 1/(n+1) and integrals?

    Have you ever see any books discussing these problems? I don't know the name of these topic.
  9. F

    Improper Integrals - Infinite Intervals

    Improper Integrals -- Infinite Intervals Homework Statement Evaluate the integral. (from e to infinity) ∫(25/x(lnx)^3)dx Homework Equations The Attempt at a Solution I know that for evaluating improper integrals, you can take the limit as t approaches infinity of the given...
  10. A

    Line integrals and paths with the same endpoints

    Homework Statement Suppose that p and q are points in U, where U is an open, path-connected, simply connected subset of Rn and c1 and c2 are smooth curves in Rn with c1(0)=c2(0)=p, c1(1)=c2(1)=q. Let w be a 1-form on U. Prove that the line integral of w over c1 equals the line integral of w...
  11. Q

    Is there someway to find the exact area of a blob using integrals?

    Someone told me Isaac Newton developed some infinitesimal triangle series to find the area of a random blob, but I think there might be some way to do it this way by drawing many lines from a central point to the edge, although that would make more of a pie slice, but is there some way to...
  12. H

    How Do Operators and Integrals Connect in Quantum Mechanics?

    Greetings chaps, This will probably be old hat to most of you, but I'm beginning to start Quantum mech. so that I can develop a deeper understanding of its application in Chemistry ( I'm a Chemistry undergrad -gauge my level from that if you will!) i.) First of all, would I be right in...
  13. L

    Convergence of a sequence of integrals

    Homework Statement Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$ converges to M The Attempt at a Solution Where do I start? I'm thinking of having g_n(x)=...
  14. GreenGoblin

    MHB How do I evaluate these tricky integrals?

    Please help me to evaluate the following integrals: 1) $\int\frac{x^{4}+1}{x^{2}+1}dx$ I recognise the form of $x^{2}+1$ in the denominator corresponds to an inverse tangent derivative. But how would I deal with the numerator in this respect? 2) $\int\frac{1}{x^{2}+x-6}dx$ I believe this...
  15. S

    Definite integrals with -infinity low bound

    I see equations of the form, y=\int_{-\infty }^{t}{F\left( x \right)}dx a lot in my texts. What exactly does it mean? From the looks of it, it just means there is effectively no lower bounds. I looked up improper integrals, but I can't say I really understand what is going on. So when...
  16. N

    Time integrals of free particle propagator

    Homework Statement As part of an assignment on matter wave diffraction I'm to calculate the following integrals I_1=\int_0^{\infty}G(\vec r_2,\vec r_1;\tau)e^{i\omega\tau}d\tau,\quad I_2=\int_0^{\infty}G(\vec r_2,\vec r_1;\tau)e^{i\omega\tau}\frac{d\tau}{\tau} Homework Equations To do so...
  17. H

    Surface Integrals: Solving Q6 & Q7

    Ok, so for Q6, I first said that z = 3 - 3x - 1.5y Using (∂z/∂x)^2 = 9, (∂z/∂y)^2 = 9/4 I then did a double integral of (x + y + (3 - 3x - 1.5y)) * sqrt(9 + 9/4 + 1) dA Letting y and x be bounded below by 0 as stated, and x bounded above by 1 - 0.5y and y bounded above by 2, I went...
  18. J

    Mastering Integrals: Tips and Tricks for Convergence and Divergence

    Been a long time I had my integral class so I forgot almost everything I knew... I need to integrate to see if the serie converge (limn→∞ an = 0). Thus, there is a theorem of the integral, if you evaluate the limit of the integral of a serie when it tends to the infinite minus when x=1 you can...
  19. H

    Computing integrals on the half line

    Hi, In my fluids work I have come to integrals of the type: \int_{0}^{\infty}\frac{e^{ikx}}{ak^{2}+bk+c}dk I was thinking of evaluating this via residue calculus but I can't think of the right contour, any suggestions? Mat
  20. polygamma

    MHB Understanding Riemann Integrals of $\ln\ x$

    $\displaystyle \int_{0}^{1} \ln \ x \ dx $ is not a proper Riemann integral since $\ln \ x $ is not bounded on $[0,1]$. Yet $ \displaystyle \int_{0}^{1} \ln \ x \ dx = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left(\frac{k}{n} \right)$. Is this because $\ln \ x$ is monotone on $(0,1]$?
  21. T

    Integrals of products of Hermite polynomials

    Hey people, I need to calculate inner product of two Harmonic oscillator eigenstates with different mass. Does anybody know where I could find a formula for \int{ H_n(x) H_m(\alpha x) dx} where H_n, H_m are Hermite polynomials?
  22. S

    Poisson integral formula to solve other integrals

    Homework Statement Use 1) \frac{1}{2\pi}\int\limits_{-\pi}^{\pi} \frac{r_0^2 - r^2}{r_0^2 - 2rr_0cos(\theta-t) + r^2} dt = 1 to compute the integral: 2) \int\limits_{-\pi}^{\pi} \left[1 - acos(x) \right]^{-1} dx for 0<a<1 [/itex]. The Attempt at a Solution I looked on Wolfram...
  23. Y

    Evaluate the integral over the helicoid [Surface integrals]

    Homework Statement Evaluate the integral \int\int_S \sqrt{1+x^{2}+y^{2}}dS where S:{ r(u,v) = (ucos(v),usin(v),v) | 0\leq u\leq 4,0\leq v\leq 4\pi } 2. The attempt at a solution Here is my attempt, I am fairly sure I am right, but it is an online assignment and it keeps telling me I am...
  24. B

    Why Does the Derivative of 2 + tan(x/2) Include a .5 Term?

    Homework Statement I understand everything except why the derivative of 2 + tan (x/2) is .5 + sec^2(x/2) I don't understand the .5 part. I understand the sec part. I would think the derivative of 2 would be C, or just disappear.
  25. polygamma

    MHB Exploring Intriguing Integrals

    Here's an eclectic bunch.1) $ \displaystyle \int_{0}^{\infty} \frac{x^{2}}{(1+x^{5})(1+x^{6})} \ dx $2) $ \displaystyle \int_{0}^{\infty} \frac{1}{(1+x^{\varphi})^{\varphi}} \ dx $ where $\varphi$ is the golden ratio3) $ \displaystyle \int_{0}^{\infty} \sin \left(x^{2} + \frac{1}{x^{2}} \right)...
  26. S

    Question regarding certain standard integrals.

    Homework Statement OK, this is something that stumped me. Homework Equations \int_{}^{}{\frac{dZ}{ A^{2}-Z^{2}}} = - \int_{}^{}{\frac{dZ}{ Z^{2}-A^{2} }} Right? \int_{}^{}{\frac{dZ}{ A^{2}-Z^{2} }}=\; \frac{1}{2A}\ln \left\{ \frac{A+Z}{A-Z} \right\}+\mbox{C} \int_{}^{}{\frac{dZ}{...
  27. B

    Apply the mean value theorem for integrals

    Homework Statement The Attempt at a Solution My book is not explaining very well the steps at solving these problems. There is a step that I'm missing step 1. find 1/(b-a) easy step 2. find the antiderivative of 4 - x, easy, x^2/2 step 3. plug in what the result of the...
  28. S

    MHB Can you prove this result for any $m,n\in\mathbb{Z}^+$?

    Prove that: \[ \int_{0}^{\pi/2} \cos(nx) \cos^n(x) dx =\frac{\pi}{2^{n+1}}\] \[ \int_{0}^{\pi} \frac{1-\cos(nx)}{1-\cos(x)} dx =n\pi \] where \( n \in \mathbb{N} \). You can use induction, contour integration or any other method you like.
  29. S

    MHB Can You Solve These Challenging Definite Integral Problems?

    Fun! Fun! Fun! Here are more entertaining problems: 1.\( \displaystyle \int_{2}^{4} \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(3+x)}+\sqrt{\ln(9-x)}}dx\) 2.\( \displaystyle \int_{\sqrt{\ln(2)}}^{\sqrt{\ln(3)}}\frac{x \sin^2(x)}{\sin(x^2)+\sin(\ln(6)-x^2)}dx\) 3.\( \displaystyle...
  30. R

    Calculate Complex Integrals with Series of Sin Z

    By parameterizing the curve (not by Cauchy's theorem) and using the series of sin z, nd the value of ∫z^k sin(z)dz around a closed Contour C where C is the unit circle z=e^(iθ), for 0≤θ<2π What do they mean by using series of sin z ? I mean if I expand it .. I get e^(iθ)- e^(3iθ)/3! --- and...
  31. Δ

    Prove Convergence of Series of Integrals | a_n |^2

    Homework Statement Let f be a continuously differentiable function on the interval [0,2\pi], where f(0) = f(2\pi) and f'(0) = f'(2\pi). For n = 1,2,3,\dotsc, define a_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) dx. Prove that the series \sum_{n=1}^\infty |a_n|^2 converges...
  32. B

    Confusion over line integrals, Green's Theoreom, Conservative fields

    Folks, 1) If we have \int F \cdot dr that is independent of the path, does that mean that the integral will always be 0? 2) For 2 dimensional problems when we evaluate line integrals directly and use Greens Theorem for every piece wise smooth closed curves C, arent we always calculating...
  33. B

    Line Integrals 2: Evaluate Triangle on Vertices (0,0), (3,3), (0,3)

    Homework Statement Evaluate this integral directly Homework Equations \int cos x sin y dx +sin x cos y dy on vertices (0,0), (3,3) and (0,3) for a triangle The Attempt at a Solution Does this have to evaluated parametrically using r(t)=(1-t)r_0+tr_1 for 0 \le t\le 1 or can I just...
  34. D

    Integrals of Expeced Valute For Normal Order Statistics

    Integrals of Expeced Value For Normal Order Statistics 1. Find the expected value of the largest order statistic in a random sample of size 4 from the standard normal distribution. Homework Equations E(X(4,4))=4∫xf(x)(F(x))^3dx, (from minus infinite to plus infinite), where f(x) is the...
  35. H

    Fraction of integrals with different variables

    how would one evaluate this without using trig substitution? Is it possible to make one integral out of this? {int[(y^2 + a1^2)^-1]dy +c1}/{int[(x^2 + a2^2)^-1]dx +c2} +c3 the numbers behind the 'a's and 'c's are supposed to be subscripts. Also, how would one deal with this...
  36. G

    Solving Non-Conservative Vector Field Line Integrals

    Hi, I'm studying calculus 3 and am currently learning about conservative vector fields. ============================= Fundamental Theorem for Line Integrals ============================= Let F be a a continuous vector field on an open connected region R in ℝ^{2} (or D in ℝ^{3}). There exists...
  37. B

    Integrals over different domains

    Folks, When we are evaluating integrals like the following, what are we evaluating in terms of units etc. For example if I integrate Fdx I get an area which represents the energy where F is the force and d is the displacement so the units are Nm etc. 1) Integrals over intervals ...
  38. E

    Path Integral Basics (Why dimension increases in the integrals?)

    Alright, I have a kind of dumb question: Why do I distinguish between dq and dqi when considering the propagation from qi to q to qf? For example, if we want the wave function at some qf and tf given qi and ti, we may write: ψ(qf,tf)=∫K(qftf;qiti)ψ(qi,ti)dqi Why do we distinguish between dqi...
  39. Y

    Solving Integrals by Series Expansions

    In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of...
  40. D

    Equality of integrals => equality of integrands

    hi folks, one often reads \int_A f(x) dx = \int_A g(x) dx for arbirary A, thus f(x) = g(x), since the equaltiy of the Integrals holds for any domain A. I don't see, why the argument "...for any domain A..." really justifies this conclusion. Can someone explain this to me, please?
  41. E

    Dirac Notation in building Path Integrals

    Alright, so I was wondering if anyone could help me figure out from one step to the next... So we have defined |qt>=exp(iHt/\hbar)|q> and we divide some interval up into pieces of duration τ Then we consider <q_{j+1}t_{j+1}|q_{j}t_{j}> =<q_{j+1}|e-iHτ/\hbar|q_{j}>...
  42. J

    Convergence of several improper integrals

    There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D...
  43. L

    Gaqussian, density cube files and exchange integrals

    Dear all, I need some help with Gaussian. I would like to know if it is possible to make cube file of the density of the single molecular orbital in Gaussian (not just overall or alpha and beta spin density) and if it is possible, wuold you be so kind to tell me how to do that? And also, is...
  44. O

    Adding two integrals with different limits of integration

    Homework Statement Interpret the integrals (from 0 to 4)∫ (3x/4) dx + (from 4 to 5)∫ (sqrt(25-x^2)) dx as areas and use the result to express the sum above as one definite integral. Evaluate the new integral. Homework Equations The Attempt at a Solution I see that I could...
  45. C

    Surface area using double integrals

    Find the surface area of the triangle with vertices (0,0) (L, L) (L,-L) I know I have to take the double integrals of f(x,y) but I have no idea what f(x,y) is supposed to be!
  46. M

    Proving & Solving Integrals with Multiplication Theorem

    Homework Statement Prove \sqrt{\frac{2}{\pi}}\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=t^{-\frac{1}{2}} and use that to solve \int^{\infty}_0\cos y^2dy Is this good way to try to prove? Homework Equations The Attempt at a Solution Homework Statement...
  47. T

    Taking Multiple Surface Integrals

    Homework Statement http://img687.imageshack.us/img687/1158/skjermbilde20111204kl85.png The Attempt at a SolutionI thought this was pretty hard and involved a number of different parts. Here's my work: Let x=cosθ and z=sinθ, also let 0≤y≤2-x=2-cosθ. I parametrize Q1, which I define to be...
  48. W

    Conceptual Questions on Line Integrals

    So we have 4 things: -Scalar Line Integral -integral of f(c(t))||c'(t)||dt from b to a -length of C: integral on curve C of ||c'(t)||dt -Vector Line Integral -integral of F(c(t))●c'(t)dt from b to a -Scalar Surface Integral -surface integral: double integral of f(Φ(u,v))||n(u,v)||dudv on...
  49. A

    Brownian Motion and Path Integrals

    I am reading "Quantum Mechanics and Path Integrals" (by Feynman and Hibbs) and working out some of the problems... as a hobby of sorts. I have run into a problem in section 12-6 Brownian Motion. On page 339 (of the emended edition), the authors demonstrate, by example, a method for...
  50. P

    Derivatives and integrals help

    derivative of integral over e^t to t^5 (sqrt(8+x^4)) dx I know I need to use the chain rule and I can take the derivative of the integral without respect to e^t and t^5. If you know the answer, can you answer and tell me how to do it?! Calculus final on Monday...
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