Fundamental Theorem of Calculus, Part 2

The Second Fundamental Theorem of Calculus (the Evaluation Theorem) states that if \(F\) is any antiderivative of a continuous \(f\) (that is, \(F'(x)=f(x)\)), then

The four panels

• Top-left - the integral: \(f(t)\) with the region from \(a\) to \(x\) shaded (teal positive, orange negative). Its signed area is the left-hand side.
• Top-right - the antiderivative \(F\): reading off \(F(a)\) and \(F(x)\); the coloured drop is the difference \(F(x)-F(a)\).
• Bottom-left - left-hand side: the traced area under the graph of \(f(t)\), that is \(\int_a^x f(t)\,dt\), revealed with a leading dot.
• Bottom-right - right-hand side: \(F(x)-F(a)\) at the same scale. The two bottom curves are identical — that is the theorem.

Any antiderivative works

Adding a constant to \(F\) leaves \(F(x)-F(a)\) unchanged, so the choice of antiderivative is irrelevant. Moving the lower limit \(a\) shifts both bottom curves by the same constant, so they stay locked together. The Fresnel integrand \(\sin(\tfrac{\pi}{2}x^2)\) has no elementary antiderivative; here \(F=S(x)=\int_0^x\sin(\tfrac{\pi}{2}t^2)\,dt\) is built by numerical integration and the theorem still holds.

Numerical method

Every integral uses composite 10-point Gauss–Legendre integration (spectral accuracy for smooth \(f\)). The left panel computes \(\int_a^x f(t)\,dt\) directly; the right panel evaluates the antiderivative difference by an independent route. They agree to many digits.