5.3 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between derivatives and integrals. It shows that these operations are inverses, allowing us to find areas under curves and solve complex problems. This powerful tool connects different calculus concepts.

By linking differentiation and integration, this theorem opens up new ways to analyze functions. It's not just about formulas – it helps us understand how things change over time, making it crucial for physics, engineering, and many other fields.

The Fundamental Theorem of Calculus

Mean Value Theorem for Integrals

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• States for a continuous function $f$ on the closed interval $[a,b]$, there exists a point $c$ in $(a,b)$ such that $\int_{a}^{b} [f(x)](https://www.fiveableKeyTerm:f(x)) dx = f(c)(b-a)$
  • Geometrically represents a rectangle with base $b-a$ and height $f(c)$ having the same area as the region under the curve $f(x)$ from $a$ to $b$
• Applies to estimating the average value of a function over an interval ($f(c)$ represents the average value)
• Proves the Fundamental Theorem of Calculus by establishing a connection between the definite integral and the function values
• Relies on the continuity of the function to ensure the existence of $c$

Fundamental Theorem of Calculus, Part 1

• States if $f$ is continuous on $[a,b]$ and $F(x) = \int_{a}^{x} f(t) dt$, then $F$ is an antiderivative of $f$ on $[a,b]$, meaning $\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$
• Evaluates the derivative of an integral function $F(x) = \int_{a}^{x} f(t) dt$ by simply taking $f(x)$
  • Example: If $F(x) = \int_{1}^{x} t^2 dt$, then $F'(x) = x^2$ (the integrand evaluated at the upper limit)
• Demonstrates that integration and differentiation are inverse processes
• Relates to the concept of accumulation, as $F(x)$ represents the accumulated area under the curve of $f(t)$ from $a$ to $x$

Applications of Fundamental Theorem, Part 2

• States if $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$
• Computes definite integrals by:
  1. Finding an antiderivative $F(x)$ of the integrand $f(x)$
  2. Evaluating $F(b) - F(a)$, where $a$ and $b$ are the lower and upper limits of integration
  • Example: To evaluate $\int_{0}^{1} x^2 dx$, find an antiderivative $\frac{1}{3}x^3 + C$, then compute $(\frac{1}{3}(1)^3 + C) - (\frac{1}{3}(0)^3 + C) = \frac{1}{3}$
• Can be used to calculate the exact value of a definite integral, which is the limit of Riemann sums as the number of subdivisions approaches infinity

Connections and Applications of the Fundamental Theorem of Calculus

Differentiation vs integration relationship

• Establishes differentiation and integration as inverse operations
• The derivative of the integral of a function is the original function (Part 1)
• The integral of a function can be computed using an antiderivative (Part 2)
• Solves various calculus problems like finding areas under curves and determining total change in a function over an interval
• Provides a method for calculating instantaneous rate of change from accumulated change

Combined Fundamental Theorem concepts

• Some problems require using both parts of the Fundamental Theorem of Calculus
  • Example: To find the area under $y = \frac{d}{dx} \int_{0}^{x} \sin(t) dt$ from $x=0$ to $x=\pi$:
    1. Use Part 1 to find $f(x) = \frac{d}{dx} \int_{0}^{x} \sin(t) dt = \sin(x)$
    2. Use Part 2 to calculate $\int_{0}^{\pi} \sin(x) dx = [-\cos(x)]_{0}^{\pi} = 2$

Connections in mathematical analysis

• Applies to various areas like differential equations, Fourier analysis, and probability theory
  • Solves certain types of differential equations
  • Develops Fourier series and transforms
  • Computes probabilities and expectations of continuous random variables
• Provides a foundation for advanced topics such as multivariable calculus, vector calculus, and complex analysis
• Shares conceptual similarities with the Fundamental Theorem of Algebra, as both theorems establish fundamental relationships in their respective fields