Mathematics

Problem

Which of the following expressions best represents an approximation of

\displaystyle \int_a^b f(x)\,dx

based on a partition

a=x_0<x_1<\dots <x_n=b

with non-uniform subinterval widths

\Delta x_i = x_i - x_{i-1}

and sample points

c_i\in[x_{i-1},x_i]

for

i=1,2,\dots ,n

? a)

\displaystyle \sum_{i=1}^{n} f(x_{i-1})\,\Delta x_i

\displaystyle \sum_{i=1}^{n} f(c_i)\,\Delta x_i

\displaystyle \sum_{i=1}^{n}\bigl(f(x_{i-1})+f(x_i)\bigr)\,\Delta x_i

\displaystyle \sum_{i=1}^{n} f(c_i)\,\dfrac{b-a}{n}