Relationship Between Definite Integrals and Series Sums

1. When a function is continuous on the closed interval :
   where and .
   
   
   
   
   This expresses the integral as the limit of a Riemann sum, which approximates the area under the curve by dividing the interval into subintervals and summing the areas of the corresponding rectangles.
   
   
2. The following methods express series as definite integrals: