#integrals — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #integrals, aggregated by home.social.
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Alright, future engineers!
**FTC:** The link between differentiation & integration.
Ex: `int[a,b] f'(x)dx = f(b) - f(a)`.
Pro-Tip: It's the bridge from rates of change to total accumulation. Crucial for definite integrals!
#Calculus #Integrals #STEM #StudyNotes -
**Antiderivative:** Reverse of differentiation. Finds the function whose derivative is given.
Ex: If `f'(x) = 2x`, then `f(x) = x^2 + C`.
Pro-Tip: ALWAYS add `+ C`! It represents all possible constant terms.
#Calculus #Integrals #STEM #StudyNotes -
Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated -
Ah, another riveting revelation that "areas" need definitions! 🤔 This mind-blowing journey through #rectangles and partitions will surely redefine your life—assuming, of course, that you've never heard of #calculus before. 🙄 Bravo for connecting #integrals and #derivatives, an insight only centuries old! 🎉
https://david.alvarezrosa.com/posts/fundamental-theorem-of-calculus/ #areas #definitions #HackerNews #ngated -
A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes -
A **Definite Integral** calculates the signed area under a curve between two points.
Ex: `int(f(x)dx)` from `a` to `b` = `F(b)-F(a)`.
Pro-Tip: Think total accumulation! It's used for total displacement, volume, or work done.
#Calculus #Integrals #STEM #StudyNotes -
Alright, future engineers!
An **Integral** calculates the total accumulation of a quantity, often representing the area under a curve. Ex: `∫[a,b] f(x) dx` is the area from `a` to `b`. Pro-Tip: Think of it as summing up infinitely tiny slices!
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Alright, future engineers!
An **Integral** calculates the total accumulation of a quantity, often representing the area under a curve. Ex: `∫[a,b] f(x) dx` is the area from `a` to `b`. Pro-Tip: Think of it as summing up infinitely tiny slices!
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Alright, future engineers!
A **Definite Integral** calculates the *net area* under a function's curve between two points. Ex: Area = ∫[a,b] f(x) dx. Pro-Tip: Think of it as summing infinitely many tiny rectangles!
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Alright, future engineers!
A **Definite Integral** calculates the *net area* under a function's curve between two points. Ex: Area = ∫[a,b] f(x) dx. Pro-Tip: Think of it as summing infinitely many tiny rectangles!
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An integral finds the *total accumulation* or area under a curve. Ex: Finding total distance traveled from a velocity function. Pro-Tip: It's essentially the *antiderivative* – reversing differentiation. Always add +C for indefinite integrals!
#Calculus #Integrals #STEM #StudyNotes -
An integral finds the *total accumulation* or area under a curve. Ex: Finding total distance traveled from a velocity function. Pro-Tip: It's essentially the *antiderivative* – reversing differentiation. Always add +C for indefinite integrals!
#Calculus #Integrals #STEM #StudyNotes -
It’s incredible that these two identities can explain so much.
Link: https://arxiv.org/abs/2505.03754
#math #calculus #euler #unification #generalization #integrals #integration
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Here's a little math problem that breaks at least one problem solver and one LLM. It breaks them in the sense that they don't know how to solve it, which is marginally better than spewing out authoritative-looking nonsense.
\int \frac{e^{x}}{\sqrt{1-16e^{2x}}} dx
See the image for my solution.
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Oh, the sheer thrill of evaluating integrals! 🙄 Let's all pretend #Feynman single-handedly invented #math while ignoring the guy named #Leibniz who did it decades earlier. 🤦♂️ Clearly, contour integration is for those who didn't get the memo on Feynman's party trick. 🎉
https://zackyzz.github.io/feynman.html #integrals #education #contourintegration #HackerNews #ngated -
Oh, the sheer thrill of evaluating integrals! 🙄 Let's all pretend #Feynman single-handedly invented #math while ignoring the guy named #Leibniz who did it decades earlier. 🤦♂️ Clearly, contour integration is for those who didn't get the memo on Feynman's party trick. 🎉
https://zackyzz.github.io/feynman.html #integrals #education #contourintegration #HackerNews #ngated -
Learning Feynman's Trick for Integrals
https://zackyzz.github.io/feynman.html
#HackerNews #FeynmanTrick #Integrals #Learning #Physics #Math #Hacks #Education
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Learning Feynman's Trick for Integrals
https://zackyzz.github.io/feynman.html
#HackerNews #FeynmanTrick #Integrals #Learning #Physics #Math #Hacks #Education
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have you ever needed a symbol for three snakes sharing the same hula hoop?
well, unicode has got you covered!
U+2230 : ∰
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have you ever needed a symbol for three snakes sharing the same hula hoop?
well, unicode has got you covered!
U+2230 : ∰
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Dubious Math in Infinite Jest (2009)
https://www.thehowlingfantods.com/dfw/dubious-math-in-infinite-jest.html
#ycombinator #theorem #odds #number #108 #integrals #mathematical #math #reading #mike #tie #39 #errors #states #match #page -
Dubious Math in Infinite Jest (2009)
https://www.thehowlingfantods.com/dfw/dubious-math-in-infinite-jest.html
#ycombinator #theorem #odds #number #108 #integrals #mathematical #math #reading #mike #tie #39 #errors #states #match #page -
Intentional math errors in David Foster Wallace's work (2009)
https://www.thehowlingfantods.com/dfw/dubious-math-in-infinite-jest.html
#ycombinator #theorem #odds #number #108 #integrals #mathematical #math #reading #mike #tie #39 #errors #states #match #page -
Intentional math errors in David Foster Wallace's work (2009)
https://www.thehowlingfantods.com/dfw/dubious-math-in-infinite-jest.html
#ycombinator #theorem #odds #number #108 #integrals #mathematical #math #reading #mike #tie #39 #errors #states #match #page -
Integrals of inverse functions!
Proof without words (see image; credit: Jonathan Steinbuch, CC BY-SA 3.0, via Wikimedia Commons)...
For any montonic and invertible function \(f(x)\) in the interval \([a,b]\):
\[\displaystyle\int_a^bf(x)~ \mathrm dx+\int_{f(a)=c}^{f(b)=d}f^{-1}(x)~\mathrm dx=b\cdot f(b)-a\cdot f(a)=bd-ac\]If \(F\) is an antiderivative of \(f\), then the antiderivatives of \(f^{-1}\) are:
\[\boxed{\displaystyle\int f^{-1}(y)~\mathrm dy=yf^{-1}(y)-F\circ f^{-1}(y)+C}\]
where \(C\) is an arbitrary constant (of integration), and \(\circ\) is the composition operator (function composition).For example:
\[\begin{align*}\displaystyle\int \sin^{-1}(y) \, \mathrm dy &= y\sin^{-1}(y) - (-\cos(\sin^{-1}(y)))+C\\ &=y\sin^{-1}(y)+\sqrt{1-y^2}+C\end{align*}\]\[\displaystyle\int \ln(y) \, dy = y\ln(y)-\exp(\ln(y)) + C= y\ln(y)-y + C.\]
#Function #InverseFunction #InverseFunctions #Functions #Integral #Integrals #Antiderivative #Integration #Calculus #FunctionComposition #CompositeFunction)
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The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.
Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters -
It's really fucking aggravating when you try to compute an integral, get to a solution, and then when you look at the instructor's solution, you find him using a formula that WAS NEVER INTRODUCED BEFORE JUST NOW!
🤬
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It's really fucking aggravating when you try to compute an integral, get to a solution, and then when you look at the instructor's solution, you find him using a formula that WAS NEVER INTRODUCED BEFORE JUST NOW!
🤬
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The USM 𝐢𝐧𝐜𝐨𝐫𝐩𝐨𝐫𝐚𝐭𝐞𝐬, 𝐞𝐱𝐭𝐞𝐧𝐝𝐬, 𝐣𝐮𝐬𝐭𝐢𝐟𝐢𝐞𝐬, and 𝐬𝐮𝐫𝐩𝐚𝐬𝐬𝐞𝐬 Euler's substitutions. In Euler's substitutions, the choice of signs based on the domain must be made manually, whereas in the USM, the supporting theorems prescribe which sign to use according to the domain. Moreover, the USM shows that Weierstrass substitutions and the use of complex exponentials for integration are merely two sides of the same coin. The USM not only 𝐮𝐧𝐢𝐟𝐢𝐞𝐬 these two techniques into one, but also 𝐠𝐞𝐧𝐞𝐫𝐚𝐥𝐢𝐳𝐞𝐬 them.
USM: https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
#math #calculus #integrals #technique #new #halfangleapproach #symmetry
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Y el último video de esta tanda de 5 videos sobre integración. Aquí resuelvo un ejemplo con substitución trigonométrica, ¡harta diversión! https://youtu.be/7TGZ4gzeTQM #calculo #matemáticas #maths #calculus #integrales #integrals
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An excellent general result.
If \(\Re(s)>1\),
\[\displaystyle\int_0^\infty\frac{\ln x}{x^s+1}~\mathrm dx=\frac{\pi^2}{4s^2}\left[\sec^2\left(\frac{\pi}{2s}\right)-\csc^2\left(\frac{\pi}{2s}\right)\right]\]Special cases:
\[\displaystyle\int_0^\infty\frac{\ln x}{x^2+1}~\mathrm dx=0\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^3+1}~\mathrm dx=-\frac{2\pi^2}{27}\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^4+1}~\mathrm dx=-\frac{\pi^2}{8\sqrt2}\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^5+1}~\mathrm dx=-\frac{4\pi^2}{25}\left(\frac{2+\sqrt5}{5+\sqrt5}\right)=-\frac{(5+3\sqrt5)\pi^2}{125}\]#Integral #Integrals #GeneralResult #GeneralResults #Result #Results #Logarithms #Logarithm #Integration #DefiniteIntegral #Calculus #IntegralCalculus
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An excellent general result.
If \(\Re(s)>1\),
\[\displaystyle\int_0^\infty\frac{\ln x}{x^s+1}~\mathrm dx=\frac{\pi^2}{4s^2}\left[\sec^2\left(\frac{\pi}{2s}\right)-\csc^2\left(\frac{\pi}{2s}\right)\right]\]Special cases:
\[\displaystyle\int_0^\infty\frac{\ln x}{x^2+1}~\mathrm dx=0\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^3+1}~\mathrm dx=-\frac{2\pi^2}{27}\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^4+1}~\mathrm dx=-\frac{\pi^2}{8\sqrt2}\]
\[\displaystyle\int_0^\infty\frac{\ln x}{x^5+1}~\mathrm dx=-\frac{4\pi^2}{25}\left(\frac{2+\sqrt5}{5+\sqrt5}\right)=-\frac{(5+3\sqrt5)\pi^2}{125}\]#Integral #Integrals #GeneralResult #GeneralResults #Result #Results #Logarithms #Logarithm #Integration #DefiniteIntegral #Calculus #IntegralCalculus
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Take a look at this interesting integral. #MathsChallenge [Hint: use the properties of the Jacobi theta function of the third type \(\vartheta_3(z,q)\)]
\[\boxed{\displaystyle\int_0^{\frac{\pi}{4}}\dfrac{1+2\displaystyle\sum_{n\geq1}e^{-n^2\pi x}}{1+2\displaystyle\sum_{n\geq1}e^{-n^2\pi/x}}\ \mathrm{d}x=\sqrt\pi}\]
#MathChallenge #IntegralChallenge #InterestingIntegral #WeirdIntegral #Integral #Integrals #DefiniteIntegral
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Integral challenge!
\[\displaystyle\int_0^\infty\ln\left(1+\dfrac{\cosh\alpha}{\cosh x}\right)\ dx=\dfrac{\pi^2}{8}-\dfrac{\arccos^2(\cosh\alpha)}{2}\]
#Integral #Integrals #IntegralChallenge #HyperbolicFunction #HyperbolicCosine #Logarithm #DefiniteIntegral
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Interesting integral! #Challenge
\[\displaystyle\int_0^1\dfrac{\ln (x)\ln(1-x)}{x(1-x)}\operatorname{Li}_2(x)\ dx=5\zeta(2)\zeta(3)-8\zeta(5)\]
Where \(\operatorname{Li}_2(x)\) denotes the dilogarithm (or Spence's function), and \(\zeta(x)\) denotes the Riemann zeta function.#ZetaFunction #Zeta #Dilogarithm #SpenceFunction #Polylogarithm #Integral #DefiniteIntegral #Integration #Integrals #RiemannZetaFunction #Logarithm #Function #LogarithmicFunction
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I also studied the same in the course of electrical engineering education.
In terms of #calculus, I think that derivatives *are* intuitive.
But #integrals?
Integrals are #CounterintuitiveMagic.
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I also studied the same in the course of electrical engineering education.
In terms of #calculus, I think that derivatives *are* intuitive.
But #integrals?
Integrals are #CounterintuitiveMagic.
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The integral
\[ \int_0^\infty x^k e^{-x} dx = k! \]
is used as motivation for the gamma function and in the irrationality proof of \( e \). But it can also be used for the transformation \( T \) defined by
\[ T f = \int_0^\infty f(x) e^{-x} dx. \]
If \( f = \sum_i f_i x^i \) is a power series and everything converges, it is transformed to \( T f = \sum_i f_i i! \). One can therefore say that \( T \) evaluates \( f \) at the factorial and write
\[ T f = f(!). \]
Are there other unusual places at which one can evaluate a function?
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The integral
\[ \int_0^\infty x^k e^{-x} dx = k! \]
is used as motivation for the gamma function and in the irrationality proof of \( e \). But it can also be used for the transformation \( T \) defined by
\[ T f = \int_0^\infty f(x) e^{-x} dx. \]
If \( f = \sum_i f_i x^i \) is a power series and everything converges, it is transformed to \( T f = \sum_i f_i i! \). One can therefore say that \( T \) evaluates \( f \) at the factorial and write
\[ T f = f(!). \]
Are there other unusual places at which one can evaluate a function?
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https://www.youtube.com/watch?v=MwVBzE7Z5gw
Huh maybe if I saw this back in 1st year Uni I wouldn't've failed Calc 1 and would've stayed in the CS program? Could've been a dev. But nah, swapped over to IS&T degree (already met its math req). Kinda was more what I wanted to do anyway though. I still sometimes do wonder.
#Integrals #Calculus #Integration -
https://www.youtube.com/watch?v=MwVBzE7Z5gw
Huh maybe if I saw this back in 1st year Uni I wouldn't've failed Calc 1 and would've stayed in the CS program? Could've been a dev. But nah, swapped over to IS&T degree (already met its math req). Kinda was more what I wanted to do anyway though. I still sometimes do wonder.
#Integrals #Calculus #Integration -
HÖLDER'S INEQUALITY:
If \(\mathcal{S}\) is a measurable subset of \(\mathbf{R}^n\) with the Lebesgue measure, and \(\eta\) and \(\xi\) are measurable real- or complex-valued functions on \(\mathcal{S}\), then\[\displaystyle\left(\int_\mathcal{S}|\eta(x)\xi(x)|\ \mathrm{d}x\right)^{ab}\leq\left(\int_\mathcal{S}|\eta(x)|^a\ \mathrm{d}x\right)^b\left(\int_\mathcal{S}|\xi(x)|^b\ \mathrm{d}x\right)^a\]
where, \((a,b)\in(1,\infty)^2\) with \(1/a+1/b=1\).
#HolderInequality #LebesgueMeasure #Integrals
