What is Limits of integration: Definition and 84 Discussions

In calculus and mathematical analysis the limits of integration of the integral





{\displaystyle \int _{a}^{b}f(x)\,dx}
of a Riemann integrable function f defined on a closed and bounded [interval] are the real numbers


{\displaystyle a}


{\displaystyle b}
. The region that is bounded can be seen as the area inside


{\displaystyle a}


{\displaystyle b}
For example, the function




{\displaystyle f(x)=x^{3}}
is bounded on the interval


{\displaystyle [2,4]}






{\displaystyle \int _{2}^{4}x^{3}\,dx}

with the limits of integration being


{\displaystyle 2}


{\displaystyle 4}

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  1. KungPeng Zhou

    A question about definite integrals and series limits

    In my opinion , if it can be shown that this is a monotonically bounded sequence, one can confirm that there is a limit. First,we know $$ \frac{1-x^{4n}}{1+x^{2}}dx=(1-x^{2}) (1+x^{2}) ^{n-1}=(1-x^{4}) ^{n-1}(1+x^{2}).$$ According to the integral median theorem,we can get $$a_n=(2- \sqrt{3} )...
  2. C

    No Limits of Integration for Electric Field Integral?

    For this problem, The solution is, However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q. However, how would you solve this problem with limits of integration? Many thanks!
  3. F

    Jacobian: how to change limits of integration?

    Hello, I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##. From there, I can reformulate with respect to ##z## and...
  4. Samama Fahim

    Limits of Integration in the Transmission Coefficient

    Initially '0' is the upper limit and ##a = \frac{Ze^2}{E}## is the lower limit. With change of variable ##x = \frac{Er}{Ze^2}##, for ##r=0##, ##x=0##, and for ##r=\frac{Ze^2}{E}##, ##x=1##, so 1 should be the lower limit. However, he takes 1 as the upper limit, and without a minus sign. Why is...
  5. agnimusayoti

    Changing Variables and the Limits of Integration using the Jacobian

    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...
  6. JD_PM

    What are the limits of integration for this surface integral?

    I want to compute: $$\oint_{c} F \cdot dr$$ I have done the following: $$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$ I don't know what limits the surface integral will have. Actually, I am not sure what's the surface. May you shed some light...
  7. opus

    Why no change in limits of integration here?

    Homework Statement Please see attached image for the full scope of the problem, and to see the work drawn out by the text. My question lies with line 3 as it is clear that u-substitution was used on a definite integral, but the limits of integration were not changed. Homework EquationsThe...
  8. Mr Davis 97

    I Changing the limits of integration, getting 0

    I have the integral ##\displaystyle \int_0^{2 \pi} \frac{1-\cos x}{3+\cos x} ~ dx##. I want to make the tangent half-angle substitution ##t = \tan (x/2)## so that I can get a rational function. However, both limits of integration just become zero. This is the first case. In the second case, I...
  9. A

    Finding the limits of integration and quadratic formula

    Homework Statement Please look at the photo! Homework Equations -x^2+4x=x^2-6x+5 The Attempt at a Solution I got 2x^2-10x+5 but it says it's wrong
  10. C

    I Limits of integration on Polar curves

    General question, how do you determine the limits of integration of a polar curve? Always found this somewhat confusing and can't seem to find a decent explanation on the internet.
  11. Dopplershift

    Limits of Integration of a Triangle

    Homework Statement Suppose you have a Triangle with the vertices, (0,0) (1,1) and (0,1). Integrating along that path. I have some differential function dZ where Z = Z(x,y) Homework EquationsThe Attempt at a Solution [/B] If I need to integrate, then I need to find the limits of...
  12. L

    Limit case of integral with exp and modified Bessel function

    Homework Statement How to integrate this? ##\int_{0}^{A} x e^{-a x^2}~ I_0(x) dx## where ##I_0## is modified Bessel function of first kind? I'm trying per partes and looking trough tables of integrals for 2 days now, and I would really really appreciate some help. This is a part of a...
  13. T

    I Integration Limits Changing in Double Integral Order Change

    For part of a proof of a differential equations equivalence, we needed to use that $$\int_0^t [\int_0^s g(\tau,\phi(\tau))\space d\tau]\space ds = \int_0^t [\int_\tau^t ds]\space g(\tau,\phi(\tau))\space d\tau$$ I understand that the order is being changed to integrate with respect to s first...
  14. qq545282501

    Volume bounded by two surfaces, what am I missing?

    Homework Statement Find the volume of the solid bounded by z=x^2+y^2 and z=8-x^2-y^2 Homework Equations use double integral dydx the textbook divided the volume into 4 parts, The Attempt at a Solution [/B] f(x)= 8-x^2-y^2-(x^2+y^2)= 4-x^2-y^2 i use wolfram and got 8 pi, the correct...
  15. Calpalned

    Limits of integration for electric potential

    In the example in the picture, we can see that they chose the limits of integration to be from 0 to R_0. Why didn't they choose x (that is, from 0 to x)? Isn't that what we normally integrate over when we find potential energy and electric fields? Thank you
  16. U

    Light-like Geodesic - What are the limits of integration?

    Homework Statement Consider the following geodesic of a massless particle where ##\alpha## is a constant: \dot r = \frac{\alpha}{a(t)^2} c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)} Homework EquationsThe Attempt at a Solution Part (a) c \frac{dt}{d\lambda} = \frac{\alpha}{a} a dt =...
  17. Yam

    Concept of Moment of Inertia and its limits of integration

    Homework Statement I am trying to work the moment of inertia for a) rotating rod, axis through the centre of the rod http://hyperphysics.phy-astr.gsu.edu/hbase/mi2.html#irod3 b) Solid cylinder http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl2 [/B] Homework Equations I = R^2 dM The...
  18. T

    Limits of integration

    Homework Statement sketch the solid region contained within the sphere, x^2+y^2+z^2=16, and outside the cone, z=4-(x^2+y^2)^0.5. b) clearly identifying the limits of integration, (using spherical coordinates) set up the iterated triple integral which would give the volume bounded by the...
  19. C

    MHB Finding new limits of integration problem

    In the integral integral(1,infinity) e^(-sqrt(x)) / sqrt(x) STEP 1: I let u = -sqrt(x) du = -1/(2sqrt(x)) then my lower bound u = -1 then my upper bound u = -infinity -2 integral(-1,infinity) e^u du I would then switch the order of the integration bounds and multiply by -1My question is...
  20. fluidistic

    Statistical mechanics, confused about an approximation and limits of integration

    Homework Statement Hello, I tried to solve a problem on my own and then I looked up a solution on the web, and I realize that it seems that I goofed. The problem statement can be found at http://www.hep.fsu.edu/~reina/courses/2012-2013/phy5524/homework/solutions/hw5_sol.pdf (Problem 1, part...
  21. A

    Changing limits of integration - definite integral (without u sub)?

    Hello Can someone please tell me how is: \int_{-R}^{R} \frac{\cos mx}{x^2 + 1}\,dx = 2\int_{0}^R \frac{\cos mx}{x^2 + 1}\,dx where, m and R are positive real numbers This is how I'm trying to solve it... \int_{-R}^R \frac{\cos mx}{x^2 + 1}\,dx = \int_{-R}^0 \frac{\cos mx}{x^2 + 1}\,dx...
  22. B

    Changing limits of integration in subsitution

    Hi, I'm doing this integration: I = ∫^{1}_{-1}1/(\pi√1-x2)dx I made the substitution x = cosθ, and I'm fine with performing the integral apart from changing the limits - for x = 1 I put θ = 0, but for x = -1, how do I know whether to choose θ = \pi or θ = -\pi? The first choice gives I...
  23. vanceEE

    Limits of Integration for Revolving Region R around the y-axis

    Homework Statement Let f be the function defined by $$ f(x) = - ln(x) for 0 < x ≤ 1. $$ R is the region between the graph of f and the x-axis. http://learn.flvs.net/webdav/educator_apcalcbc_v10/module08/imgmod08/08_10_01.gif b. Determine whether the solid generated by revolving region R...
  24. J

    Limits of Integration Variable

    Knowing that the limits of integration of a any function, for example: \int_{-\infty}^{+\infty}\delta (x)dx=1 I know that's correct call your primitive through the limit superior as a variable, so H(x)=\int_{-\infty}^{x}\delta (x)dx But, and if I want to describe your primitive through the...
  25. I

    Limits of integration question (double integral)

    so in the image in the link below, i don't understand a couple of things: 1.) the center of the cylinder is off to the side and not at the center. where/how in the problem are we taking this into account? because it should definitely affect the volume under the parabaloid right? 2.) most of...
  26. Q

    Finding Bounds of Integration for Integral of Cos(2+x)

    Homework Statement http://i.minus.com/jJQzZXoxXFqEB.png Homework Equations (b-a)/n = Δx The Attempt at a Solution I know how to express the sum as an integral .. almost. It is the integral of cos(2+x) with respect to x. However, what are my bounds of integration? I know that b-a must...
  27. I

    Expressing the limits of integration for radius in polar coordinates

    i'm trying to integrate some some function bounded by the x-y domain of x2+y2=6y which is a circle on the x-y plane shifted upward where the outer part of the circle is 6. i'm trying to integrate a double integral.. ∫∫f(x)rdrdθ i don't know how to express my limits of integration for r...
  28. C

    Both limits of integration change to zero.

    Homework Statement Integrate (1 + x2)1/2 from -∏ to ∏ Homework Equations The Attempt at a Solution I substiuted x = tan(theta) but when I went to change the limits of integration I got 0 and 0. What am I doing wrong?
  29. I

    Why is this legal? (limits of integration)

    i have one expression, and then i will expand that expression into another expression like so... y=f(x) ------> f(x)= x+5u , u= another variable now i will integrate both sides.. like so... ∫f(x)dx=∫(x+5u)du now, one fundamental rule in algebra is that, whatever is done to one...
  30. G

    Triple integral, limits of integration

    Pretty general question. Integrate f(x,y,z) dxdydz over the area defined by: x^{2} + y^{2} + z^{2} \leq 4 x \leq 0 y \leq 0 z \leq 0 It is immidiately apparent that it is 1/8 of a sphere with r=2. So from that geometrical intuition we can do a variable substitution to spherical...
  31. U

    Simple Volume Integral, limits of integration

    Homework Statement Homework Equations The Attempt at a Solution Are my limits of integration right?
  32. G

    Function of random variable, limits of integration

    Homework Statement X is uniformly distributed over [-1,1]. Compute the density function f(y) of Y = 2X2 + 1. Homework Equations The Attempt at a Solution FY(Y) = P(Y < y) = P(2X2 + 1 < y) = P(X < +\sqrt{1/2(y-1)} = FX(+\sqrt{1/2(y-1)}) We have that f(x) = 0.5 for -1 < x <...
  33. D

    Setting up limits of integration for multiple integral

    Homework Statement I need to find the volume of the region bounded by (x-1)^2 + y^2 =1 \ \ \text{and} \ \ x^2+y^2+z^2=4 \ . But I only need help setting up the limits of integration. Homework Equations The typical cylindrical change of variables. The Attempt at a Solution I have 0 \leq...
  34. P

    Comparison Theorem and Limits of integration

    Homework Statement Why is it that when using the comparison theorem my limits of integration must be from a constant value to infinity and not from negative infinity to infinity? For example ∫ x/(1+x^2) dx from -∞ to ∞
  35. N

    Improper integral, infinite limits of integration

    ∞ ∫ x/(x^2+1) dx -∞ I basicaly evaluated the integral and Ln (x^2+1) as the antiderivative and when taking the limits I get ∞-∞ (ln |1| -ln|b+1|) + (ln|n+1|- ln|1|) lim b-> neg. infinity lim n-> infinity does this function converge or diverge? this was a question on...
  36. E

    Optimizing Limits of Integration for Change of Variables

    Hi guys, I've been on quite a random change of variables binge lately and I've been messing around with a particular scenario in which I'm not 100% sure of how I should choose my limits of integration. Any help would be greatly appreciated! (And no, this is not homework, etc.) The scenario is as...
  37. H

    Reversal of limits of integration in the derivation of probability current density

    In working out the derivation of the probability current density, I see (based on the definition of j(x,t)) that the limits of integration are changed from d/dt∫(b to a) P(x.t) dx = iħ/2m[ψ*(x.t)∂/∂xψ(x.t) - ψ(x.t)∂/∂xψ*(x.t)](b to a) to d/dt∫(b to a) P(x.t) dx =...
  38. B

    Surface and Volume Integrals - Limits of Integration

    So I am trying to understand how and why the limits of surface and volume integrals come about. I think I came up with a easy to understand argument but not a mathematically sound one. Frankly its a little dodgy. Can anyone provide feedback on this argument or provide a better and possibly more...
  39. M

    Why Does Reversing Limits of Integration Affect the Answer?

    My limits of integration for my angle I chose to be from pi to 0 then I got the negative answer from what was in the book.. Shouldn't that be correct because we are integrating from -3 to 3?
  40. E

    Problem with limits of integration - converting double integral to polar form

    Homework Statement \int_0^2 \int_0^\sqrt{2x-x^2} xy,dy,dx I know the answer, but how does the 2 in the outer integral become pi/2?? I'm fine with everything else, I just can't get this...
  41. T

    Limits of Integration for Joint Distribution problems

    Suppose that (X,Y) is uniformly distributed over the regiondefined by 0≤ y ≤ 1-x2 and -1≤ x ≤ 1. a) find the marginal densities of X and Y Attempted solution: So first I have to find the joint density function which ends up being fxy(x,y) = 3/4 and then from that I would solve...
  42. S

    Spherical limits of integration for a region bounded by a cone and a praboloid

    Hi everybody, I am trying to solve the following problem and I get stuck on the last question. I would appreciate a lot that someone helps me . Here is the problem: Let D be the region bounded from below by the cone z= the root of (x^2 + z^2), and from above by the paraboloid z = 2 – x^2 –...
  43. B

    Double Integral with Strange Limits of Integration

    Homework Statement For the double integral ∫[0,1]∫[0,x^3] e^(y/x) dxdy (a) sketch the region of integration (b) evaluate the integral and (c) re-express the integral with the order of integration reversedHomework Equations NoneThe Attempt at a Solution The problem is that I've never seen a...
  44. V

    Limits of integration for regions between polar curves

    Alright. I completely confused about determining the area between regions of polar curves. However, I do feel that I have a solid grasp in finding areas for single functions. For a given function in polar form, I know that I find the limits of integration by setting the function equal to zero...
  45. T

    Substitution changes limits of integration?

    I'm working through rewriting the gamma function as an infinite product, but my question is just about a specific substitution that was made in my textbook. They took the equation: \Gamma_ n(z)=\int_0^n t^{z-1} (1-\frac{t}{n}) ^ndt for Re(z)>0 and n greater than or equal to 1. and made the...
  46. J

    Fourier series of complex numbers with diffrent limits of integration?

    Fourier series of complex numbers with diffrent limits of integration? Dear all, i don't know how to simplify a COMPLEX NUMBER Fourier series with LIMITS OF INTEGRATION that are not complementary. I MEAN limits LIKE this X to -X being easy to solve and SIMPLIFY but Not X to -Y or...
  47. 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...
  48. jinksys

    Changing the limits of integration

    ! SOLVED ! This isn't a homework problem, just an equation in my chapter. I don't see how the two integrals pointed to by the blue arrows become the integral pointed to by the red arrow. I know that if you swap the limits of integration, you change the sign of the integral. However, how do...
  49. Z

    How to determine a Limits of Integration of Wave Packet

    Homework Statement Consider a force-free particle of mass m described, at an instant of time t = 0, by the following wave packet: \begin{array}{l} 0 \ \mathrm{for} \ |x| > a + \epsilon \\ A \ \mathrm{for} \ |x| ≤ a \\ -\frac{A}{\epsilon} (x − a − \epsilon) \ \mathrm{for} \ a < x ≤ a + \epsilon...
  50. F

    How to find the new limits of integration

    Homework Statement I'm confused about a certain aspect of shell method, washer method, and disk method. If I want to rotate using the washer method about the x-axis how do I get the new limits of integration? For example: y = 9-x2 over [0,3] about x-axis. I solved for x and got: x =...