The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is the bridge that connects the two main branches of calculus: differentiation (the study of local change) and integration (the study of global accumulation). Remarkably, it shows that these two processes are inverses of each other.

Part 1: The Derivative of an Integral

Let $f$ be a continuous function on $[a,b]$. Define a new function $F$ by: $F(x)={\int }_a^{x}f(t)dt$ for $x\in [a,b]$. The first part of the FTC states that $F$ is continuous on $[a,b]$, differentiable on $(a,b)$, and its derivative is: $F^{\prime }(x)=f(x)$

Proof Sketch

To find $F^{\prime }(x)$, we look at the difference quotient: $\frac{F(x+h)-F(x)}{h}=\frac{1}{h}{\int }_x^{x+h}f(t)dt$ By the Mean Value Theorem for Integrals, there exists a $c_h\in [x,x+h]$ such that: $\frac{1}{h}{\int }_x^{x+h}f(t)dt=f(c_h)$ As $h\to 0$, $c_h$ is squeezed toward $x$. Since $f$ is continuous, $f(c_h)\to f(x)$. Thus, $F^{\prime }(x)=f(x)$.

Part 2: The Integral of a Derivative

Let $f$ be a function such that $f^{\prime }$ is Riemann integrable on $[a,b]$. Then: ${\int }_a^b{f}^{\prime }(x)dx=f(b)-f(a)$

This part of the theorem gives us a powerful tool for evaluating definite integrals without having to compute Riemann sums. If we can find an antiderivative $F$ such that $F^{\prime }=f$, then ${\int }_a^{b}f(x)dx=F(b)-F(a)$.

Proof Sketch

Let $P=\{x_0,x_1,\dots ,x_n\}$ be any partition of $[a,b]$. By the Mean Value Theorem applied to $f$ on each subinterval $[x_{i-1},x_i]$, there exists $c_i$ such that: $f(x_i)-f(x_{i-1})=f^{\prime }(c_i)\Delta x_i$ Summing these up: ${\sum }_{i=1}^n(f(x_i)-f(x_{i-1}))={\sum }_{i=1}^n{f}^{\prime }(c_i)\Delta x_i$ The left side is a telescoping sum that equals $f(b)-f(a)$. The right side is a Riemann sum for $f^{\prime }$. As the mesh size of the partition goes to zero, the sum converges to the integral ${\int }_a^b{f}^{\prime }(x)dx$.

Leibniz’s Rule

A common application of the FTC in advanced analysis is differentiating an integral where the limits depend on a variable: $\frac{d}{dx}{\int }_{a(x)}^{b(x)}f(t)dt=f(b(x))b^{\prime }(x)-f(a(x))a^{\prime }(x)$ This is essentially a combination of the FTC Part 1 and the Chain Rule.

Change of Variables and Integration by Parts

The FTC allows us to derive the two most important tools for integration:

1. Substitution (Change of Variables): Derived from the Chain Rule. If $u=g(x)$, then $\int f(g(x))g^{\prime }(x)dx=\int f(u)du$.
2. Integration by Parts: Derived from the Product Rule. $\int udv=uv-\int vdu$.

Python: Verifying the FTC

We can use symbolic math in Python to demonstrate that differentiating the integral of a function returns the original function.

import sympy as sp

x, t = sp.symbols('x t')
f = sp.sin(t)**2 * sp.exp(t)

# Define the integral function F(x) = integral of f from 0 to x
F = sp.integrate(f, (t, 0, x))

# Differentiate F(x)
dF_dx = sp.diff(F, x)

print(f"Original f(x): {f.subs(t, x)}")
print(f"Derivative of Integral: {sp.simplify(dF_dx)}")

# They should be identical
assert sp.simplify(dF_dx - f.subs(t, x)) == 0

Summary

The Fundamental Theorem of Calculus is one of the most beautiful results in mathematics. It demonstrates a hidden symmetry between the slope of a curve and the area beneath it. Without this theorem, calculus would remain a collection of disconnected tricks; with it, it becomes a unified and powerful system for understanding the continuous world.

Exercise