Summary of Integration Formulas and the Net Change Theorem

Essential Concepts

  • The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.
  • The area under an even function over a symmetric interval can be calculated by doubling the area over the positive [latex]x[/latex]-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.

Key Equations

  • Net Change Theorem
    [latex]F(b)=F(a)+{\int }_{a}^{b}F\text{‘}(x)dx[/latex] or [latex]{\displaystyle\int }_{a}^{b}F\text{‘}(x)dx=F(b)-F(a)[/latex]


net change theorem
if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity