# What is Limits of integration: Definition and 84 Discussions

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

a

b

f
(
x
)

d
x

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

a

{\displaystyle a}
and

b

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

a

{\displaystyle a}
and

b

{\displaystyle b}
.
For example, the function

f
(
x
)
=

x

3

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

[
2
,
4
]

{\displaystyle [2,4]}

2

4

x

3

d
x

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

with the limits of integration being

2

{\displaystyle 2}
and

4

{\displaystyle 4}
.

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24. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### Simple Volume Integral, limits of integration

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