Math Portfolio

Fundamental Theorem of Calculus

The Bridge Between Differentiation and Integration

The Fundamental Theorem of Calculus is one of the most important results in mathematics, establishing the relationship between differentiation and integration. It consists of two parts:

Part 1: The Derivative of an Integral

If $f$ is a continuous function on $[a, b]$ and we define a new function $F$ by:

$$F(x) = \int_{a}^{x} f(t) \, dt$$

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

$$F'(x) = \frac{d}{dx}\left[\int_{a}^{x} f(t) \, dt\right] = f(x)$$

This means that if we take the derivative of the accumulated area function, we get back the original function.

Part 2: The Evaluation Theorem

If $f$ is a continuous function on $[a, b]$ and $F$ is any antiderivative of $f$ (that is, $F' = f$), then:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

This is often written using the notation: $\int_{a}^{b} f(x) \, dx = [F(x)]_{a}^{b} = F(b) - F(a)$

The Fundamental Theorem of Calculus provides a practical method for evaluating definite integrals without using Riemann sums directly. It connects the concept of the definite integral as the limit of Riemann sums with the concept of the antiderivative.

Examples and Applications

Example 1: Evaluating a Definite Integral

Calculate $\int_{0}^{3} x^2 \, dx$

Step 1: Find an antiderivative of $f(x) = x^2$. We have $F(x) = \frac{x^3}{3}$.

Step 2: Apply the Fundamental Theorem:

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

Example 2: Working with Part 1

If $g(x) = \int_{1}^{x} \sqrt{t} \, dt$, find $g'(4)$.

Using Part 1 of the Fundamental Theorem, we have:

$$g'(x) = \sqrt{x}$$

Therefore, $g'(4) = \sqrt{4} = 2$.

Real-world Applications

  • Finding the total distance traveled from velocity functions
  • Calculating accumulated growth from rate functions
  • Computing probability distributions in statistics
  • Determining work done by a force along a path in physics
  • Analyzing electric potential in circuit theory

Notes and Resources

Key Takeaways

  • The Fundamental Theorem of Calculus connects differentiation and integration
  • Part 1: The derivative of an accumulated area function gives the original function
  • Part 2: Definite integrals can be evaluated using antiderivatives
  • This theorem transforms calculus from theoretical to practical
  • Understanding this theorem provides insight into why techniques like the "power rule" for integration work