What is lower Riemann sum?

Question

Isaiah Allen · Accepted Answer

See below

The lower Riemann sum for the integral # int_a^b f(x) dx # involves (some authors call the result of the summation the lower Riemann sum, the limit being called the lower Riemann integral) We can define the upper Riemann sum in a similar fashion - but with #f_i# standing for the maximum value of the function #f(x)# in the interval #[x_{i-1},x_i)# For the integral to exist, both the lower and the upper Riemann sum must exist and be equal - their common value is the Riemann integral.

Emilia Aguilar · Answer

undefined A lower Riemann sum is a method for approximating the area under a curve using rectangles. It involves dividing the interval over which the function is being integrated into subintervals and then constructing rectangles based on the minimum function value within each subinterval. The height of each rectangle is determined by the minimum function value within the corresponding subinterval. The sum of the areas of these rectangles gives an approximation of the total area under the curve. As the number of subintervals increases, the lower Riemann sum approaches the actual area under the curve (the integral) if the function is Riemann integrable.

Emma Aldridge · Answer

undefined The lower Riemann sum is a way to approximate the area under a curve using rectangles. It is obtained by partitioning the interval over which the function is defined into subintervals, then constructing rectangles whose heights are determined by the minimum value of the function within each subinterval. The sum of the areas of these rectangles gives an approximation to the area under the curve. As the number of subintervals increases, this approximation becomes more accurate, converging to the actual area under the curve.