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1.4 Integration Formulas and the Net Change Theorem

3 min read•june 24, 2024

Integration formulas are the building blocks of calculus, helping us solve complex problems with ease. These formulas, like the and , allow us to find antiderivatives and calculate areas under curves.

The connects rates of change to total change over time. It's super useful for real-world applications, like finding distance traveled from or population growth from birth rates.

Integration Formulas

Basic integration formulas

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Top images from around the web for Basic integration formulas

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  • Power Rule applies ∫xn[dx](https://www.fiveableKeyTerm:dx)=xn+1n+1+C\int x^n [dx](https://www.fiveableKeyTerm:dx) = \frac{x^{n+1}}{n+1} + C where n≠−1n \neq -1 to solve problems like ∫x3dx=x44+C\int x^3 dx = \frac{x^4}{4} + C
  • multiplies a constant cc by the integral of f(x)f(x) as ∫cf(x)dx=c∫f(x)dx\int cf(x) dx = c \int f(x) dx to solve problems like ∫3x2dx=3∫x2dx=3⋅x33+C=x3+C\int 3x^2 dx = 3 \int x^2 dx = 3 \cdot \frac{x^3}{3} + C = x^3 + C
  • Sum Rule adds the integrals of f(x)f(x) and g(x)g(x) as ∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx to solve problems like ∫(x2+3x)dx=∫x2dx+∫3xdx=x33+3x22+C\int (x^2 + 3x) dx = \int x^2 dx + \int 3x dx = \frac{x^3}{3} + \frac{3x^2}{2} + C
  • subtracts the integral of g(x)g(x) from the integral of f(x)f(x) as ∫[f(x)−g(x)]dx=∫f(x)dx−∫g(x)dx\int [f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx to solve problems like ∫(x2−3x)dx=∫x2dx−∫3xdx=x33−3x22+C\int (x^2 - 3x) dx = \int x^2 dx - \int 3x dx = \frac{x^3}{3} - \frac{3x^2}{2} + C

Odd vs even function integrals

  • satisfy f(−x)=−f(x)f(-x) = -f(x) and have the property ∫−aaf(x)dx=0\int_{-a}^a f(x) dx = 0 where a>0a > 0, so ∫−22x3dx=0\int_{-2}^2 x^3 dx = 0 since x3x^3 is an
  • satisfy f(−x)=f(x)f(-x) = f(x) and have the property ∫−aaf(x)dx=2∫0af(x)dx\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx where a>0a > 0, so ∫−11x2dx=2∫01x2dx=2⋅13=23\int_{-1}^1 x^2 dx = 2 \int_0^1 x^2 dx = 2 \cdot \frac{1}{3} = \frac{2}{3} since x2x^2 is an

Definite integrals and area under a curve

  • A represents the signed area between a function and the x-axis over a specific interval
  • For , the can be interpreted as the area under the curve
  • The states that for a continuous function f(x) on [a,b], there exists a c in [a,b] such that f(c)=1b−a∫abf(x)dxf(c) = \frac{1}{b-a}\int_a^b f(x)dx

Net Change Theorem

Net change theorem applications

  • States the integral of a over [a,b][a, b] equals the net change of the quantity over that interval, expressed as ∫abdQdtdt=Q(b)−Q(a)\int_a^b \frac{dQ}{dt} dt = Q(b) - Q(a) where Q(t)Q(t) is the quantity as a function of time tt
  • Applies to scenarios like:
    1. Calculating distance traveled by an object given its (cars, projectiles)
    2. Determining total change in population given a (bacteria, cities)
    3. Computing accumulated interest earned given an (savings accounts, loans)
  • Can be used to solve certain by finding the of the rate of change function

Integrals for net change calculations

  • Distance traveled formula ∫abv(t)dt=s(b)−s(a)\int_a^b v(t) dt = s(b) - s(a) uses velocity function v(t)v(t) and position function s(t)s(t), so if v(t)=3t2v(t) = 3t^2, the distance traveled from t=1t = 1 to t=3t = 3 is ∫133t2dt=t3∣13=27−1=26\int_1^3 3t^2 dt = t^3 \bigg|_1^3 = 27 - 1 = 26
  • Population change formula ∫abr(t)dt=P(b)−P(a)\int_a^b r(t) dt = P(b) - P(a) uses population growth rate function r(t)r(t) and population function P(t)P(t), so if r(t)=100e0.02tr(t) = 100e^{0.02t}, the population change from t=0t = 0 to t=10t = 10 is ∫010100e0.02tdt=5000e0.02t∣010≈1103\int_0^{10} 100e^{0.02t} dt = 5000e^{0.02t} \bigg|_0^{10} \approx 1103
  • Accumulated interest formula ∫abi(t)dt=A(b)−A(a)\int_a^b i(t) dt = A(b) - A(a) uses interest rate function i(t)i(t) and account balance function A(t)A(t), so if i(t)=0.05i(t) = 0.05, the accumulated interest from t=0t = 0 to t=5t = 5 on an initial balance of 1000is1000 is \int_0^5 0.05 \cdot 1000 dt = 50t \bigg|_0^5 = 250$

Key Terms to Review (39)

Accumulation Function: The accumulation function, also known as the antiderivative or indefinite integral, is a fundamental concept in calculus that represents the cumulative change of a function over an interval. It describes the accumulated value of a function as it is integrated or summed up over a range of input values.
Antiderivative: An antiderivative, also known as a primitive function or indefinite integral, is a function whose derivative is the original function. It represents the accumulation or the reverse process of differentiation, allowing us to find the function that was differentiated to obtain a given derivative.
Area Under a Curve: The area under a curve represents the integral of a function over a specified interval, essentially calculating the accumulation of quantities represented by that function. This concept is crucial in understanding how to approximate and compute areas, connecting the graphical representation of functions with their corresponding numerical values. It also serves as a foundation for various mathematical applications, such as finding total distance traveled, work done, or other quantities derived from a function's rate of change.
Bald eagle: The bald eagle is the national bird of the United States and a species of eagle found in North America. It is known for its white head, brown body, and strong, hooked beak.
Constant Multiple Rule: The Constant Multiple Rule is a fundamental concept in calculus that allows for the simplification of integration by treating a constant factor outside of the integral as a multiplier. This rule is particularly useful in the context of integration formulas and the net change theorem.
Continuous Functions: Continuous functions are a fundamental concept in calculus, representing functions that change smoothly and without any abrupt jumps or breaks. Continuity is a crucial property that allows for the application of powerful integration techniques and the analysis of the net change of a function over an interval.
Deceleration: Deceleration is the rate at which an object slows down. It is represented as a negative acceleration in physics and mathematics.
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.
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.
Difference Rule: The Difference Rule is a fundamental principle in calculus that allows for the differentiation of the difference between two functions. It states that the derivative of a difference of two functions is equal to the difference of their derivatives. This rule connects to the broader principles of integration formulas and the Net Change Theorem, helping in understanding how changes in one function relate to changes in another through integration and differentiation.
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.
Displacement: Displacement is the net change in position of an object, calculated as the integral of velocity over a given time interval. It can be represented mathematically as $\int_{a}^{b} v(t) \ dt$.
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.
Even function: An even function is a function $f(x)$ such that $f(x) = f(-x)$ for all $x$ in its domain. Graphically, even functions are symmetrical with respect to the y-axis.
Even Functions: An even function is a mathematical function where the value of the function at a point is equal to the value of the function at the negative of that point. In other words, the graph of an even function is symmetric about the y-axis.
Exponential Functions: Exponential functions are mathematical expressions of the form $$f(x) = a imes b^{x}$$ where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions model rapid growth or decay and are essential in various applications, such as compound interest and population growth, due to their unique property where the rate of change is proportional to the function's current value.
Federal income tax: Federal income tax is a tax levied by the United States Internal Revenue Service (IRS) on the annual earnings of individuals, corporations, trusts, and other legal entities. It is calculated based on the amount of taxable income earned during a calendar year.
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.
Iceboat: An iceboat is a vessel designed for moving swiftly over ice, typically using sails and skates. It is primarily used in regions with frozen lakes or rivers during winter.
Indefinite integral: An indefinite integral represents the collection of all antiderivatives of a function, essentially reversing the process of differentiation. It is expressed in the form $$\int f(x) \, dx = F(x) + C$$, where $$F(x)$$ is the antiderivative of $$f(x)$$, and $$C$$ is a constant of integration that accounts for the fact that there are infinitely many antiderivatives differing only by a constant. Understanding indefinite integrals is crucial in various mathematical contexts, as they provide foundational techniques for solving equations and analyzing areas under curves.
Integration by Parts: Integration by parts is a technique used to integrate products of functions by transforming the integral into a simpler form using the formula $$\int u \, dv = uv - \int v \, du$$. This method connects various integration strategies, making it especially useful in situations where other techniques like substitution may not be effective.
Interest Rate Function: The interest rate function is a mathematical representation that describes how the interest rate varies over time, often used to model the accumulation of interest on investments or loans. This function is significant because it helps in understanding the relationship between the principal amount, time, and the total interest accrued, and it plays a crucial role in evaluating financial growth through integration. Furthermore, this concept connects with the net change theorem, as it allows for calculating the total change in value over a specified period based on varying rates of interest.
Linearity of Integrals: Linearity of integrals is a fundamental property that states the integral of a linear combination of functions is equal to the linear combination of their individual integrals. This property simplifies the evaluation of integrals and is crucial in the context of integration formulas and the net change theorem.
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.
Net change theorem: The Net Change Theorem states that the integral of a rate of change function over an interval gives the net change in the quantity over that interval. It is essentially the Fundamental Theorem of Calculus applied to real-world problems involving rates of change.
Net Change Theorem: The Net Change Theorem states that the net change in a quantity over an interval is equal to the integral of the rate of change of that quantity over that interval. This theorem connects the concepts of integration and accumulation, showing how the total accumulation of a quantity can be found by integrating its instantaneous rate of change, represented mathematically as $$ ext{Net Change} = ext{Final Value} - ext{Initial Value} = \\int_{a}^{b} f'(x) \, dx$$.
Odd function: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, it is symmetric with respect to the origin.
Odd Functions: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of the function. In other words, the graph of an odd function is symmetric about the origin, meaning that it is reflected across both the $x$-axis and the $y$-axis.
Population growth rate function: The population growth rate function describes how the number of individuals in a population changes over time, often modeled as a differential equation. It can be represented mathematically to determine the rate at which a population increases or decreases, influenced by factors like birth rates, death rates, immigration, and emigration. This function connects to integration formulas and the net change theorem by illustrating how cumulative changes in population size can be determined over a specific interval.
Power Rule: The power rule is a fundamental integration technique in calculus that allows for the efficient evaluation of integrals involving variables raised to a power. It provides a straightforward method to integrate functions of the form $x^n$, where $n$ is any real number, by applying a simple formula.
Rate of Change Function: The rate of change function, also known as the derivative function, represents the instantaneous rate of change of a function at a given point. It describes how the value of a function changes with respect to changes in its input variable.
Riemann: Riemann is a fundamental concept in calculus that refers to the mathematical framework developed by the German mathematician Bernhard Riemann for defining and analyzing integrals. This concept is central to understanding the topics of approximating areas, the definite integral, integration formulas, and improper integrals in calculus.
Sum Rule: The Sum Rule in calculus states that the integral of a sum of functions is equal to the sum of the integrals of those functions. It provides a convenient way to evaluate integrals involving multiple terms by breaking them down into simpler, individual integrals.
The Integral Symbol (∫): The integral symbol (∫) represents the mathematical operation of integration, which is the inverse of differentiation. It is used to calculate the accumulated change of a function over an interval, finding the area under a curve, or determining the total effect of a varying quantity.
Tour de France: Tour de France is a prestigious multi-stage bicycle race held primarily in France, covering various terrains and spanning over three weeks. Each stage of the race can be analyzed for distance, speed, and elevation changes using integration techniques.
Trigonometric Functions: Trigonometric functions are mathematical functions that describe the relationship between the angles and sides of a right triangle. They are fundamental in the study of calculus and are essential in understanding integration formulas, substitution, and other integration strategies.
U-substitution: U-substitution is a technique used in integration that simplifies the process by substituting a part of the integral with a new variable, usually denoted as 'u'. This method allows for easier integration by transforming complex expressions into simpler ones, facilitating the calculation of definite and indefinite integrals.
Velocity: Velocity is a vector quantity that measures the rate of change of an object's position with respect to time. It has both magnitude and direction.
Velocity Function: The velocity function, denoted as $v(t)$, represents the rate of change of an object's position with respect to time. It is a fundamental concept in calculus that is closely related to the concepts of integration formulas and the net change theorem.
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1.5 Substitution