Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [bounds-of-integration]

In many questions the problem of determining bounds of integration in multiple integrals is a major part of what an answer needs to deal with, and in surprisingly many questions it is the only issue. This tag is for such occasions.

Common topics include:

  • Switching the order of iterated integrals to find new bounds
  • Determining the bounds of a region after performing a change of variables
  • Classifying regions of different dimensions when applying one of Green's, Stokes's, or Divergence Theorems
  • Recovering the limits of integration from the passage of a Riemann sum to a definite integral
127 questions
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When is $\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$?

I would like to find positive, distinct, algebraic real numbers $a,b\in \mathbb R^+\cap\mathbb A$ satisfying $$\int_a^b \frac{1}{x}\ln\bigg(\frac{x^3+1}{x^2+1}\bigg)dx=0$$ Does anyone know of a systematic way to go about solving this problem?…
Franklin Pezzuti Dyer
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4 answers

Double Integral $\int\limits_0^1\!\!\int\limits_0^1\frac{(xy)^s}{\sqrt{-\log(xy)}}\,dx\,dy$

Is it possible to get a closed form of the following integral? $$I=\int_0^1\!\!\!\int_0^1\frac{(xy)^s}{\sqrt{-\log(xy)}}\,dx\,dy\quad\quad\quad(s>0).$$ My attempt: I’ve tried a change of variables from cartesian coordinates to polar…
user279934
  • 121
11
votes
3 answers

Is it necessary to write limits for a substituted integral?

To solve the following integral, one can use u-substitution: $$\int_2^3 \frac{9}{\sqrt[4]{x-2}} \,dx,$$ With $u = \sqrt[4]{x-2}$, our bounds become 0 and 1 respectively. Thus, we end up with the following: $$36\int_0^1{u^2} \,du$$ In the first case,…
Lord Kanelsnegle
  • 121
10
votes
6 answers

How to set the limits of a definite integral by substitution?

A worked example in my calculus textbook is to evaluate $$\int_0^\pi\sqrt{1-\sin2x}\text dx.$$ The book's approach is to use trig identities, but my idea is to use a substitution by letting $$t=1-\sin2x,$$ but I don't know how to set the limits. I…
Zirui Wang
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6
votes
4 answers

Change of variables Double integral

I have $$\iint_A \frac{1}{(x^2+y^2)^2}\,dx\,dy.$$ $A$ is bounded by the conditions $x^2 + y^2 \leq 1$ and $x+y \geq 1$. I initially thought to make the switch the polar coordinates, but the line $x+y=1$ is making it hard to find the limits of…
tamefoxes
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1 answer

When does integration via u-substitution break down, equal limits of integration?

Edited: changed $\displaystyle\int_{a}^{b}f(g(t))g'(t) \, dt = \int_{g(a)}^{g(b)}f(x) \, dx$ TO $\displaystyle\int_{a}^{b}f(t) \, dt = \int_{f(a)}^{f(b)}u \, \frac{du}{f'(f^{-1}(u))}$ I am wondering about this idea in general, but for concreteness…
bRost03
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1 answer

How can you simplify/verify this solution for $\int\limits_0^{.25991…} Q^{-1}(x,x,x)dx?$

As I do not know the complex behavior of this function, it would be even harder to integrate past the real domain. The upper bound for the domain is a constant I will denote β. $${{Q_2}=\int_0^βQ^{-1}(x,x,x)dx= \int_0^β…
Тyma Gaidash
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5
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3 answers

Is it correct to write $\int_a^x f(x) dx$?

The question pretty much sums it all. A few days ago when studying how to find the real part of a function knowing the imaginary part (or vice versa) I was given this formula: $$u(x, y) =\int_{x_0}^{x} \frac{\partial u} {\partial x} (x, y_0) \,dx +…
user556151
4
votes
5 answers

Bounds for the Harmonic k-th partial sum.

I need to bound the k-th partial sum or the Harmonic series. i.e. $$ln(k+1)<\sum_{m=1}^{k}\frac{1}{m}<1+ln(k)$$ I'm triying to integrate in $[m,m+1]$ in the relation $\frac{1}{m+1}<\frac{1}{x}<\frac{1}{m}$ for all $x\in[m,m+1]$ and I…
Ragnar1204
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4
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1 answer

How to calculate the floor integral $\int_0^{\pi}\lfloor\pi^2\cos^3x\rfloor\sin x\,dx$?

$$\int_0^{\pi}\lfloor\pi^2\cos^3x\rfloor\sin x\,dx$$ (where $\lfloor x \rfloor $ is the floor of $x$) I thought of breaking into required bounds but its too lengthy. Moreover I had to take cube root and then $\cos$ inverse. Please give a hint.
user354545
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3
votes
2 answers

Confusion in bounds while substituting $x=\sin^2\theta$ in $\int_{0}^{1}\frac{x^{2}}{\sqrt{x(1-x)}}\mathrm dx$

I really need help in evaluating the following integral from MIT-BEE Quarterfinals $2025$. The question is to evaluate the integral $$\int_{0}^{1}\frac{x^{2}}{\sqrt{x(1-x)}}\mathrm dx$$ Now I evaluated this given integral in the following way : I…
Dev
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3
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1 answer

Setting bounds for triple integral for volume between 3 surfaces

Let V be the bound region between the surfaces $x^2 = 2z, x^2 +y^2 =8z, z=6$ ($V$ is the inner part of $x^2 =z$) Find the volume of the region V. I can't seem to set the bounds for the integral. so far I've found that (1) and (2) intersect along the…
Luke
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1 answer

How do I find the bounds of this particular integral?

I want to convert this integral to Polar Coordinates to solve it: $\int_{0}^{2}\int_{0}^{\sqrt{y}}4xy^{2} \, dx \, dy$ What would be the bounds of $r$ and $\theta$ be? I know how to solve the integral and all, and I also know how to generally find…
proof-of-correctness
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3
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Calculus: Finding Volume with Triple Integrals

**Problem:**A shape is bounded by the following elliptical function $4x^2 + y^2 +z = 128$ and the planes $x=0, x=4, y=0, y=4$. Find the volume of the shape. My attempt: $4x^2 + y^2 +z = 128 \implies z = 128-4x^2-y^2$,…
AtKin
  • 608
3
votes
3 answers

Help me understand easy (not for me) concepts in volume integral

Keep looking at the page for an hour. Still not sure how I can get the sloping surface of $x+y+z=1$ and integration ranges for $x, y, z$. and why their range is different too. The book keeps throwing things at me without much explanation. Help me.
user808793
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