What is the Fundamental Theorem of Calculus Part 2 (FTC2)?

📚 What is the Fundamental Theorem of Calculus Part 2 (FTC2)?

The Fundamental Theorem of Calculus Part 2 (FTC2) provides a method for evaluating definite integrals. It states that if you know an antiderivative of a function, you can compute the definite integral of that function by evaluating the antiderivative at the upper and lower limits of integration and subtracting the values. In simpler terms, it connects differentiation and integration, showing they are inverse processes.

📜 History and Background

The FTC is not attributed to a single mathematician but developed over time, with contributions from Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. The formalization and rigorous proof came later. The FTC's development was crucial to the advancement of calculus, providing a systematic way to solve many problems related to areas, volumes, and rates of change.

🔑 Key Principles of FTC2

• 🔍 Antiderivatives: An antiderivative $F(x)$ of a function $f(x)$ is a function such that $F'(x) = f(x)$. Finding the antiderivative is the first step in applying FTC2.
• 💡 Definite Integrals: A definite integral $\int_{a}^{b} f(x) \, dx$ represents the signed area under the curve of $f(x)$ from $x=a$ to $x=b$.
• 📝 The Theorem: If $f(x)$ is continuous on the interval $[a, b]$, and $F(x)$ is an antiderivative of $f(x)$, then $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$.
• ➕ Constant of Integration: When finding an antiderivative, we usually add a constant $C$. However, when using FTC2, the constant cancels out when evaluating $F(b) - F(a)$, so it's typically omitted.

🌍 Real-World Examples

Here are a couple of examples of how FTC2 can be applied:

1. Calculating Distance from Velocity:

   Suppose a car's velocity is given by $v(t) = 3t^2 + 2t$ (in meters per second). To find the distance traveled between $t=1$ and $t=3$ seconds, we integrate the velocity function:

   $\int_{1}^{3} (3t^2 + 2t) \, dt$

   The antiderivative of $v(t)$ is $s(t) = t^3 + t^2$. Applying FTC2:

   $s(3) - s(1) = (3^3 + 3^2) - (1^3 + 1^2) = (27 + 9) - (1 + 1) = 36 - 2 = 34$ meters.
2. Calculating Change in Population:

   Suppose the rate of population growth in a city is modeled by $P'(t) = 100t + 50$ (people per year). To find the population increase from year 0 to year 5, we integrate the rate function:

   $\int_{0}^{5} (100t + 50) \, dt$

   The antiderivative of $P'(t)$ is $P(t) = 50t^2 + 50t$. Applying FTC2:

   $P(5) - P(0) = (50(5)^2 + 50(5)) - (50(0)^2 + 50(0)) = (1250 + 250) - 0 = 1500$ people.

📝 Conclusion

The Fundamental Theorem of Calculus Part 2 is a cornerstone of calculus, providing a direct method to evaluate definite integrals using antiderivatives. Its applications span various fields, making it an essential tool for solving real-world problems involving accumulation and change.