Fundamental Theorem of Calculus, Part 1

Slide \(x\) to accumulate signed area under \(f\); the accumulation \(F(x)=\int_a^x f(t)\,dt\) has slope exactly \(f(x)\). Move \(a\) and watch \(F\) shift up or down while its derivative never changes.

\[\frac{d}{dx}\int_a^x f(t)\,dt \;=\; f(x)\]
Integrand \(f\)
e.g. x^2-2, x*sin(x), exp(-x^2), 1/(1+x^2)
-10.00
\(x\)
-10.000
\(F(x)=\int_a^x f(t)\,dt\)
0.00000
\(F'(x)\)
0.00000
\(f(x)\)
0.00000
\(\displaystyle\int_a^x f(t)\,dt\) — signed area
\(F(x)=\displaystyle\int_a^x f(t)\,dt\) — accumulation
\(f(x)\) — the integrand (compare, exactly, with \(F'(x)\))
\(F'(x)=\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt\) — derivative traces out
Cite this tool 
Kapita, S. (2026). Fundamental Theorem of Calculus, Part 1. Math Tools. https://shelvean.github.io/math-tools/ftc1.html