5.4 Integration Formulas and the Net Change Theorem

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

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• 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 ($[0, a]$)

Fundamental Theorem of Calculus and Related Concepts

• 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 $t = 3$)