13.3 Fundamental theorem of calculus and early applications

The Fundamental Theorem of Calculus is a game-changer in math. It links differentiation and integration, showing they're opposite operations. This connection lets us solve complex problems by breaking them down into simpler parts.

Integration helps us calculate areas, volumes, and other accumulations. It's super useful in physics, engineering, and economics. We can find work done by forces, analyze fluid pressure, and even optimize profits using these powerful tools.

Fundamental Theorem of Calculus

Connecting Integration and Differentiation

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• Fundamental Theorem of Calculus establishes the relationship between differentiation and integration
• States that integration and differentiation are inverse operations
• Consists of two parts: FTC Part 1 and FTC Part 2
• FTC Part 1 defines the integral function $F(x) = \int_a^x f(t) dt$ and proves it is an antiderivative of f(x)
• FTC Part 2 provides a method for evaluating definite integrals using antiderivatives
• Expresses the definite integral as the difference of antiderivative values: $\int_a^b f(x) dx = F(b) - F(a)$
• Revolutionized calculus by unifying previously separate concepts

Understanding Integration

• Integration calculates the accumulation of quantities over an interval
• Represented by the integral symbol ∫, introduced by Leibniz
• Definite integral computes the signed area between a function and the x-axis
• Indefinite integral refers to the family of antiderivatives of a function
• Fundamental theorem connects definite integrals to antiderivatives
• Integration techniques include u-substitution, integration by parts, and partial fractions
• Applications range from physics (work, energy) to economics (consumer surplus)

Exploring Differentiation and Antiderivatives

• Differentiation measures the rate of change of a function at a point
• Derivative of a function f(x) denoted as f'(x) or $\frac{d}{dx}f(x)$
• Basic differentiation rules include power rule, product rule, and chain rule
• Antiderivative F(x) of a function f(x) satisfies $F'(x) = f(x)$
• Finding antiderivatives reverses the process of differentiation
• Antiderivatives are not unique, differ by a constant C
• General form of an antiderivative: $F(x) + C$, where C is the constant of integration
• Antiderivatives crucial for solving differential equations and evaluating definite integrals

Applications of Integration

Calculating Areas and Volumes

• Area under a curve computed using definite integrals
• Formula for area: $A = \int_a^b f(x) dx$, where f(x) ≥ 0 on [a,b]
• Areas between curves found by integrating the difference of functions
• Volume of solids of revolution calculated using the washer or shell method
• Washer method for rotating around x-axis: $V = \pi \int_a^b [R(x)^2 - r(x)^2] dx$
• Shell method for rotating around y-axis: $V = 2\pi \int_a^b xf(x) dx$
• Applications in geometry, engineering, and manufacturing

Physics and Engineering Applications

• Work calculated as the integral of force over distance: $W = \int_a^b F(x) dx$
• Fluid pressure and force determined using integration
• Center of mass found through integration for non-uniform objects
• Moment of inertia computed to analyze rotational motion
• Electric field and potential energy in electromagnetism utilize integration
• Heat transfer and thermodynamics employ integral calculus
• Stress and strain analysis in materials science relies on integration techniques

Optimization and Rate of Change Problems

• Optimization problems find maximum or minimum values of functions
• Process involves finding critical points and analyzing behavior
• Applications in economics (profit maximization, cost minimization)
• Used in engineering for designing efficient structures and systems
• Rate of change problems analyze how quantities change over time
• Instantaneous rate of change given by the derivative at a point
• Average rate of change over an interval computed using definite integrals
• Applications in population growth, chemical reactions, and financial modeling