[Verse 1] When you pile up tiny pieces, slice by slice The accumulation grows before your eyes But here's the secret hiding in the math Take the derivative and trace it backwards back The integral from a to x of f of t Creates a function, call it G of x, you see Now differentiate this G you've just defined The original f comes flooding back to mind [Chorus] It's the great reversal, the mirror dance Building up and breaking down in perfect balance Derivative of integral brings you home Integration, differentiation, two sides of one coin d over dx of integral a to x Equals f of x, no complex tricks Part one of the theorem, crystal clear The rate reveals what accumulation hid here [Verse 2] But wait, there's more to this elegant design Part two will blow your calculus-trained mind When limits bound your integral tight From a to b, a different insight Find any antiderivative F Where F prime equals your f, nothing less Then F of b minus F of a Gives you the area, the integral's way [Chorus] It's the great reversal, the mirror dance Building up and breaking down in perfect balance Integral a to b of f of x Equals F of b minus F of a, direct No Riemann sums with infinite parts Just plug and chug, the algebraic arts Part two transforms the impossible task Into simple substitution, that's all we ask [Bridge] Before this theorem, every curve's embrace Required limits racing through infinite space Rectangles shrinking, sums without end Now antiderivatives extend their hand The hardest problems become routine When inverse operations bridge the scene What seemed like magic now makes perfect sense The fundamental truth, mathematically dense [Outro] So remember when the integrals seem tough The fundamental theorem's powerful stuff Differentiation and integration dance Forever linked in calculus romance
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