#integrals — Public Fediverse posts

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

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

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)