Analysis vs Numerical Analysis

• In mathematics, solve a problem through equations; algebra, calculus, differential equations (DE), Partial DE, $\ldots$
• In numerical analysis; four operations (add, substract, multiply, division) and Comparision.
  • These operations are exactly those that computers can do
    
    $$\int_0^\pi \sqrt{1+cos^2x}dx$$
    
    
    length of one arch of the curve y-sinx; no solution with " a substitution" or "integration by parts"
    numerical analysis can compute the length of this curve by standardized methods that apply to essentially any integrand
• Another difference between a numerical results and analytical answer is that the former is always an approximation
  • this can usually be as accurate as needed (level of accuracy)