The Fundamental Theorem of Calculus (FTC) is one of the most important theorems in mathematics, connecting the two main operations of calculus: differentiation and integration. It has two parts:

**Part 1:** If $f$ is continuous on the closed interval $[a, b]$, then the function $F$ defined by

$$F(x) = \int_a^x f(t) \, dt$$

is an antiderivative of $f$ on $[a, b]$.

**Part 2:** If $f$ is continuous on the closed interval $[a, b]$ and $F$ is an antiderivative of $f$ on $[a, b]$, then

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

**Proof of Part 1:**

Let $x$ and $x + h$ be any two points in the interval $(a, b)$. Then, by the Mean Value Theorem, there exists a point $c$ in the interval $(x, x + h)$ such that

$$F(x + h) - F(x) = f(c)h$$

Dividing both sides by $h$, we get

$$\frac{F(x + h) - F(x)}{h} = f(c)$$

Taking the limit of both sides as $h \to 0$, we get

$$\lim_{h \to 0} \frac{F(x + h) - F(x)}{h} = \lim_{h \to 0} f(c)$$

But the limit on the left-hand side is $F'(x)$, by definition of the derivative. And the limit on the right-hand side is $f(x)$, since $f$ is continuous at $x$. Therefore, we have

$$F'(x) = f(x)$$

This shows that $F$ is an antiderivative of $f$ on $[a, b]$.

**Proof of Part 2:**

Let $F$ be an antiderivative of $f$ on $[a, b]$. Then, by the Mean Value Theorem, there exists a point $c$ in the interval $(a, b)$ such that

$$F(b) - F(a) = f(c)(b - a)$$

Therefore,

$$\int_a^b f(x) \, dx = F(b) - F(a) = f(c)(b - a)$$

This shows that the value of the definite integral $\int_a^b f(x) \, dx$ is equal to the area under the curve of $f$ over the interval $[a, b]$.

**Example:**

Let $f(x) = x^2$. Then, an antiderivative of $f$ is $F(x) = \frac{x^3}{3}$. Therefore, by the FTC, we have

$$\int_0^3 x^2 \, dx = F(3) - F(0) = \frac{27}{3} - 0 = 9$$

This is consistent with the fact that the area under the curve of $f(x) = x^2$ over the interval $[0, 3]$ is equal to 9.

The FTC is a powerful tool that can be used to solve a wide variety of problems in calculus and mathematics. It is also used in many other fields, such as physics, engineering, and economics.
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