[Verse 1] Picture a curve drawn on a grid Area trapped beneath where it lives Cut it in slices, paper thin Rectangles standing, let's begin Each little bar has width delta x Height matches function where it rests Multiply them, stack them high Watch the magic multiply [Chorus] Integral from a to b Riemann sums in harmony Limit as n approaches infinity Sigma f of x-i-star times delta x precisely Drop by drop fills up the sea Moment by moment, you and me The miracle of accumulation Infinite slices, finite destination [Verse 2] Velocity curves give distance traveled Force times displacement, work unraveled Probability density weighs the chance Every application gets its dance Delta x equals b minus a Divided by n, that's the law As rectangles grow thinner still The limit bends to our will [Chorus] Integral from a to b Riemann sums in harmony Limit as n approaches infinity Sigma f of x-i-star times delta x precisely Drop by drop fills up the sea Moment by moment, you and me The miracle of accumulation Infinite slices, finite destination [Bridge] Leibniz knew what he was drawing That elongated S keeps calling "Sum it up!" the symbol cries "f of x dx" never lies From discrete to continuous flow Watch those rectangles shrink and grow Until they vanish into space Leaving area in their place [Chorus] Integral from a to b Riemann sums in harmony Limit as n approaches infinity Sigma f of x-i-star times delta x precisely Drop by drop fills up the sea Moment by moment, you and me The miracle of accumulation Infinite slices, finite destination [Outro] Continuous functions guarantee This limit will exist, you see From tiny pieces, something whole Integration feeds the soul
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