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Calculus II
Table of Contents
Unit 1 Overview: Integration
1.1 Approximating Areas
1.2 The Definite Integral
1.3 The Fundamental Theorem of Calculus
1.4 Integration Formulas and the Net Change Theorem
1.5 Substitution
1.6 Integrals Involving Exponential and Logarithmic Functions
1.7 Integrals Resulting in Inverse Trigonometric Functions

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1.3 The Fundamental Theorem of Calculus

Citation:

The Fundamental Theorem of Calculus connects differentiation and integration, showing they're inverse operations. It proves that finding the area under a curve and calculating rates of change are related, revolutionizing how we solve math problems.

This theorem has two parts. The first shows how to find antiderivatives, while the second gives us a way to evaluate definite integrals. Together, they provide powerful tools for solving real-world problems in physics, economics, and engineering.

The Fundamental Theorem of Calculus

Mean Value Theorem for Integrals

  • States for a continuous function $f$ on $[a, b]$, there exists a point $c$ in $[a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$
  • Geometrically means the area under the curve $f(x)$ over $[a, b]$ equals the area of a rectangle with base $b - a$ and height $f(c)$ (average value of $f$ on $[a, b]$)
  • Guarantees the existence of an average value for a continuous function over a closed interval (mean value)
  • Provides a connection between the definite integral and the average value of a function (area and average height)
  • Relies on the intermediate value theorem to ensure the existence of point $c$

Fundamental Theorem of Calculus

  • Part 1 (Existence of antiderivatives) states if $f$ is continuous on $[a, b]$, then the function $F(x) = \int_a^x f(t) dt$ is an antiderivative of $f$ on $[a, b]$, meaning $F'(x) = f(x)$
  • Part 2 (Evaluating definite integrals) states if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$, providing a method for evaluating definite integrals using antiderivatives
  • Connects the concepts of differentiation and integration, showing they are inverse operations (derivative of integral is original function, integral of derivative is original function up to a constant)
  • Demonstrates the definite integral of a function's derivative over an interval equals the change in the function's value over that interval (rate of change and accumulated change)
  • Also known as the Newton-Leibniz formula

Derivatives of integrals

  • To find the derivative of a definite integral with a variable upper limit, use $\frac{d}{dx} \int_a^x f(t) dt = f(x)$, where $a$ is a constant and $f$ is continuous on $[a, x]$ (rate of change of accumulated area)
  • To find the derivative of a definite integral with a variable lower limit, use $\frac{d}{dx} \int_x^b f(t) dt = -f(x)$, where $b$ is a constant and $f$ is continuous on $[x, b]$ (negative rate of change of accumulated area)
  • Allows for finding the rate of change of a quantity defined by an integral (marginal cost, marginal revenue)
  • Can be applied to accumulation functions to analyze their behavior

Computing definite integrals

  • To evaluate a definite integral $\int_a^b f(x) dx$:
    1. Find an antiderivative $F(x)$ of $f(x)$
    2. Compute $F(b) - F(a)$ (fundamental theorem of calculus part 2)
  • Often easier than using Riemann sums or other approximation techniques (trapezoidal rule, Simpson's rule)
  • Useful for finding areas, volumes, and other accumulated quantities (work, average value)

Differentiation vs integration

  • Differentiation finds the rate of change or slope of a function at a point (instantaneous rate of change)
  • Integration finds the accumulated value or area under a curve over an interval (total change)
  • The fundamental theorem of calculus connects these two concepts, showing they are inverse operations
    • The derivative of the integral of a function is the original function (FTC part 1)
    • The definite integral of a function's derivative over an interval equals the total change in the function's value over that interval (FTC part 2)
  • Understanding the relationship between differentiation and integration is crucial for solving various problems in calculus and its applications (optimization, differential equations)

Continuity and Limits in the Fundamental Theorem

  • Continuity of the function is a key requirement for the Fundamental Theorem of Calculus
  • The theorem relies on the properties of continuous functions, including the ability to take limits
  • The chain rule is often used in conjunction with the Fundamental Theorem when dealing with composite functions

Key Terms to Review (36)

Aphelion: Aphelion is the point in the orbit of a celestial body where it is farthest from the Sun. It is crucial for understanding variations in orbital mechanics and energy computations.
Definite integral: The definite integral of a function between two points provides the net area under the curve from one point to the other. It is represented by the integral symbol with upper and lower limits.
Evaluation theorem: The Evaluation Theorem is a key part of the Fundamental Theorem of Calculus. It states that the definite integral of a function over an interval $[a, b]$ can be found using its antiderivative.
Fundamental Theorem of Calculus, Part 1: The Fundamental Theorem of Calculus, Part 1 states that if $F$ is an antiderivative of $f$ on an interval $[a, b]$, then the integral of $f$ from $a$ to any point $x$ in that interval is equal to $F(x) - F(a)$. It links the process of differentiation and integration.
Fundamental Theorem of Calculus, Part 2: The Fundamental Theorem of Calculus, Part 2 states that if $F$ is an antiderivative of a continuous function $f$ on an interval $[a, b]$, then the integral of $f$ from $a$ to $b$ is given by $F(b) - F(a)$. It connects differentiation with integration, showing that these two operations are essentially inverses of each other.
Integration by parts: Integration by parts is a technique used to integrate the product of two functions. It is based on the product rule for differentiation and is expressed as $$ \int u \, dv = uv - \int v \, du $$.
Newton: Sir Isaac Newton was an English mathematician, physicist, astronomer, and author who is widely recognized as one of the most influential scientists of all time. In calculus, his work laid the groundwork for differential and integral calculus through the development of early concepts.
Perihelion: Perihelion is the point in the orbit of a planet, asteroid, or comet where it is closest to the Sun. Calculations involving perihelion often require integral calculus to determine orbital parameters and distances.
Rate of change: Rate of change quantifies how one quantity changes with respect to another. In calculus, it is often represented as the derivative of a function.
Skydiver: A skydiver is an individual who jumps from an aircraft and falls freely before deploying a parachute to slow their descent. The motion and forces experienced by a skydiver can be analyzed using calculus, particularly through integration.
Wingsuits: Wingsuits are specialized jumpsuits designed to allow a person to glide through the air using fabric flaps between the arms and legs. They are used in extreme sports for aerial navigation before deploying a parachute.
Trapezoidal Rule: The trapezoidal rule is a numerical integration method used to approximate the definite integral of a function over a given interval. It is a simple and widely-used technique for estimating the area under a curve by dividing the interval into trapezoids and summing their areas.
Simpson's Rule: Simpson's rule is a numerical integration method used to approximate the definite integral of a function over an interval. It is a more accurate alternative to the Riemann sum method, particularly for functions that can be well approximated by quadratic polynomials.
Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus is a central result in calculus that establishes a deep connection between the concepts of differentiation and integration. It provides a powerful tool for evaluating definite integrals and understanding the relationship between the rate of change of a function and the function itself.
Continuity: Continuity is a fundamental concept in calculus that describes the smooth and uninterrupted behavior of a function. It is a crucial property that allows for the application of various calculus techniques, such as differentiation and integration, to analyze the behavior of functions.
Mean Value Theorem for Integrals: The Mean Value Theorem for Integrals states that if a function $f(x)$ is continuous on a closed interval $[a, b]$, then there exists at least one point $c$ in the interval such that the value of the integral of $f(x)$ over the interval $[a, b]$ is equal to the product of the length of the interval and the value of the function at the point $c$. This theorem provides a way to approximate the average value of a function over an interval.
Definite Integral: The definite integral represents the area under a curve on a graph over a specific interval. It is a fundamental concept in calculus that allows for the quantification of the accumulation of a quantity over a given range.
Antiderivatives: An antiderivative, also known as a primitive function or indefinite integral, is a function whose derivative is the original function. It represents the reverse operation of differentiation, allowing one to find a function that has a given derivative.
Continuous Function: A continuous function is a function that has no abrupt changes or jumps in its graph. It is a function where small changes in the input result in small changes in the output, with no sudden or drastic variations. Continuity is a fundamental concept in calculus and is crucial for understanding the Fundamental Theorem of Calculus.
Differentiation: Differentiation is a fundamental concept in calculus that describes the rate of change of a function at a specific point. It involves finding the derivative, which represents the slope or tangent line to the function at that point, and provides insights into the behavior and properties of the function.
Integration: Integration is a fundamental concept in calculus that represents the inverse operation of differentiation. It is used to find the area under a curve, the volume of a three-dimensional object, and other important quantities in mathematics and science.
Rate of Change: The rate of change is a measure of how a quantity changes over time or with respect to another variable. It describes the speed or velocity at which a change occurs, and is a fundamental concept in calculus that underpins the understanding of derivatives and integrals.
Derivative: The derivative is a fundamental concept in calculus that measures the rate of change of a function at a particular point. It represents the slope of the tangent line to the function at that point, providing information about the function's behavior and how it is changing.
Accumulated Change: Accumulated change refers to the total or cumulative change that occurs over a given interval or period of time. It is a fundamental concept in calculus, particularly in the context of the Fundamental Theorem of Calculus, which establishes the relationship between the accumulation of change and the rate of change.
Optimization: Optimization is the process of finding the best solution or outcome among a set of available alternatives, typically by maximizing desirable outcomes or minimizing undesirable ones. It is a fundamental concept in mathematics, engineering, economics, and decision-making, where the goal is to identify the most efficient or optimal way to achieve a specific objective or target.
Limits: Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a particular value. They are used to analyze the properties of functions, such as continuity and differentiability, and form the foundation for the Fundamental Theorem of Calculus.
Dx: The term 'dx' represents an infinitesimally small change or increment in the independent variable 'x' within the context of integral calculus. It is a fundamental concept that connects the definite integral, the Fundamental Theorem of Calculus, integration formulas, inverse trigonometric functions, areas between curves, and various integration strategies.
Differential Equations: Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model and analyze various phenomena in science, engineering, and other fields where the rate of change of a quantity is of interest.
Accumulation Functions: Accumulation functions, also known as antiderivatives or indefinite integrals, are functions that represent the accumulation or running total of a given rate of change or derivative function. They are a crucial concept in the study of calculus, particularly in the context of the Fundamental Theorem of Calculus.
Intermediate Value Theorem: The Intermediate Value Theorem states that if a continuous function takes on two different values, then it must also take on all values in between those two values. It is a fundamental result in calculus that helps establish the existence of solutions to certain equations.
Chain Rule: The chain rule is a fundamental concept in calculus that allows for the differentiation of composite functions. It provides a systematic way to find the derivative of a function that is composed of other functions.
Newton-Leibniz Formula: The Newton-Leibniz formula is a fundamental theorem in calculus that connects the concepts of differentiation and integration. It states that the definite integral of a function can be evaluated by finding the antiderivative (or indefinite integral) of that function.
$f(x)$: $f(x)$ is a mathematical function that represents a relationship between an independent variable $x$ and a dependent variable $y$. The function assigns a unique output value $y$ to each input value $x$, allowing for the modeling and analysis of various phenomena and relationships in mathematics, science, and engineering.
$\int$: $\int$ represents the integral symbol, used in calculus to denote the process of integration, which is the summation of infinitely many infinitesimal quantities. It connects the concept of antiderivatives to the area under curves, allowing us to find total accumulation and evaluate the fundamental relationship between differentiation and integration.
Integral: An integral is a fundamental concept in calculus that represents the accumulation of quantities, often interpreted as the area under a curve defined by a function. Integrals can be classified into definite integrals, which calculate the total accumulation over an interval, and indefinite integrals, which represent families of functions whose derivatives yield the original function. This concept is crucial for connecting various mathematical ideas, such as area, volume, and the relationship between differentiation and integration.
Instantaneous Rate of Change: The instantaneous rate of change refers to the rate at which a function is changing at a specific point, essentially capturing how fast the value of the function is changing at that instant. This concept is crucial for understanding the behavior of functions and is mathematically represented by the derivative. It connects to various fundamental ideas, such as limits, and serves as a foundational element in calculus, particularly when discussing the relationship between differentiation and integration.