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Calculus I
Table of Contents
Unit 5 Overview: Integration
5.1 Approximating Areas
5.2 The Definite Integral
5.3 The Fundamental Theorem of Calculus
5.4 Integration Formulas and the Net Change Theorem
5.5 Substitution
5.6 Integrals Involving Exponential and Logarithmic Functions
5.7 Integrals Resulting in Inverse Trigonometric Functions

∫calculus i review

5.4 Integration Formulas and the Net Change Theorem

Citation:

Integration formulas are essential tools for solving calculus problems. They provide shortcuts for finding antiderivatives and calculating definite integrals. Understanding these formulas helps simplify complex integration tasks and solve real-world problems.

The Fundamental Theorem of Calculus bridges differentiation and integration, showing how they're inverse operations. This connection allows us to calculate definite integrals using antiderivatives and solve problems involving accumulation and net change over time.

Integration Formulas

Basic integration formulas

  • Power Rule applies the formula $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ to integrate expressions with variable exponents ($x^3$)
  • Constant Multiple Rule multiplies a constant factor outside the integral by the result of integrating the expression inside the integral ($3x^2$)
  • Sum Rule breaks the integral of a sum into the sum of separate integrals ($x^2 + 3x$)
  • Difference Rule breaks the integral of a difference into the difference of separate integrals ($x^2 - 3x$)

Integrals of odd vs even functions

  • Odd functions have the property $f(-x) = -f(x)$
    • Integrating an odd function ($x^3$) over a symmetric interval ($[-a, a]$) always equals zero
  • Even functions have the property $f(-x) = f(x)$
    • Integrating an even function ($x^2$) over a symmetric interval ($[-a, a]$) equals twice the integral over half the interval ($[0, a]$)
  • The Fundamental Theorem of Calculus connects differentiation and integration
  • Definite integral represents the area under the curve of a function over a specific interval
  • Indefinite integral is the antiderivative of a function, representing a family of functions
  • Accumulation function measures the accumulated area under a curve from a fixed point to a variable upper limit

Net Change Theorem

Net change theorem interpretation

  • The Net Change Theorem relates the integral of a rate of change ($\frac{dQ}{dt}$) over an interval ($[a, b]$) to the net change in the quantity ($Q(b) - Q(a)$)
  • Applies to various quantities that change over time such as distance traveled ($s(t)$), population ($P(t)$), and volume ($V(t)$)

Applications of net change theorem

  • Solve applied problems by:
    1. Identifying the rate of change function (velocity $v(t) = 3t^2 + 2t$)
    2. Setting up the integral using the Net Change Theorem ($\int_1^3 (3t^2 + 2t) dt = s(3) - s(1)$)
    3. Evaluating the integral ($\int_1^3 (3t^2 + 2t) dt = 34$)
    4. Interpreting the result in context (car traveled 34 meters from $t = 1$ to $t = 3$)

Key Terms to Review (9)

Bald eagle: The bald eagle is a bird of prey found in North America, known for its white head and tail feathers. It is the national bird and symbol of the United States.
Displacement: Displacement is the net change in position of an object, calculated as the difference between the final and initial positions. It can be determined using definite integrals in calculus.
Even function: An even function is a function $f(x)$ that satisfies the condition $f(-x) = f(x)$ for all $x$ in its domain. Graphically, even functions are symmetric with respect to the y-axis.
Federal income tax: Federal income tax is a tax levied by the United States government on the annual earnings of individuals, corporations, trusts, and other legal entities. It is calculated based on a progressive tax system with different rates applied to different income brackets.
Iceboat: An iceboat is a vehicle designed to travel over ice using sails for propulsion. It operates on the principles of wind force and friction reduction, allowing it to achieve high speeds.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. The graph of an odd function is symmetric about the origin.
Net change theorem: The Net Change Theorem states that the integral of a rate of change over an interval gives the net change in the quantity over that interval. Mathematically, it is expressed as $\int_{a}^{b} F'(x) \, dx = F(b) - F(a)$.
Tour de France: The Tour de France is an annual men's multiple stage bicycle race primarily held in France. It is one of the most prestigious and difficult cycling events in the world, covering approximately 3,500 kilometers over three weeks.
Velocity: Velocity is the rate of change of position with respect to time. It is a vector quantity, meaning it has both magnitude and direction.