The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) establishes the relationship between differentiation and integration, two of the main operations in calculus. It consists of two parts.

Part 1: The Derivative of an Integral

If $f$ is continuous on $[a, b]$, and we define:

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

Then $F$ is differentiable on $(a, b)$, and:

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

Intuition

This part tells us that if we integrate a function and then differentiate the result, we get back the original function. Integration and differentiation are inverse operations!

Part 2: Evaluating Definite Integrals

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then:

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

Example

Let's evaluate $\int_0^2 x^2,dx$

1. Find an antiderivative: $F(x) = \frac{x^3}{3}$
2. Apply FTC Part 2:

$$\int_0^2 x^2,dx = F(2) - F(0) = \frac{8}{3} - 0 = \frac{8}{3}$$

Why It Matters

The FTC is arguably the most important theorem in calculus because it:

1. Connects two major operations: Shows that differentiation and integration are inverse processes
2. Enables calculation: Provides a practical method to evaluate definite integrals
3. Has wide applications: Used throughout physics, engineering, economics, and more

Applications

Area Under a Curve

The definite integral $\int_a^b f(x),dx$ gives the net area between $f(x)$ and the x-axis from $x=a$ to $x=b$.

Physics: Work and Energy

Work done by a variable force $F(x)$:

$$W = \int_a^b F(x),dx$$

Probability

For a probability density function $f(x)$:

$$P(a \leq X \leq b) = \int_a^b f(x),dx$$

Conclusion

The Fundamental Theorem of Calculus is a cornerstone of mathematical analysis. It not only provides computational tools but also deepens our understanding of the relationship between rates of change and accumulation.