Math 11B- Calculus

Math 11B- Calculus

1. Approximating and computing areas

   • Sigma Notation
     • The Greek capital letter Σ, sigma, is used to express long sums of values in a compact form.
     • Ex) Without Sigma Notation: 1+2+3+4+5+...+19+20 With Sigma Notation: Σ i=1 20 Ai
   • • By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area.
   • Reinmann Sum
     • We could evaluate the function at any point Ci in the subinterval [Xi-1, Xi], and use f(xi) as the height of our rectangle

2. The Definite Intergral

   • Definition: Generalizes the concept of the area under a curve. We lift the requirements that f(x) be continuous and nonnegative
   • Net Signed Area
     • Taking the limit as n approaching infinity, the Riemann sum approaches the area between the curve above the x-axis and the x-axis, less the area between the curve below the x-axis and the x-axis
   • Total Area

3. The Indefinite Integral

   • Can be defined as the family of antiderivatives for a function. We use the notation of the integral sign without any limits,

4. Fundamental Theorem of Calculus

5. Net Change

   • Can be applied to area, distance, and volume. It accounts for negative quantities

6. Substitution

   • We substitute part of the integrand with the variable "u" and part of the integrand with "du". It is also referred to as change of variables because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.

7. Area Between the Curve

   • We will look back on Reinmann sum
   • Area of compound regions

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