Split up [1, 3] into 4 subintervals:
[1, 3/2], [3/2, 2], [2, 5/2], [5/2, 3]
each with length (3 - 1)/2 = 1/2.
The right endpoints
are {3/2, 2, 5/2, 3}, which we can index by the sequence
![r_i=1+\dfrac i2](https://tex.z-dn.net/?f=r_i%3D1%2B%5Cdfrac%20i2)
with
.
Evaluating the function at the right endpoints gives the sampling points
, {27/4, 5, 11/4, 0}.
Then the area is approximated by
![\displaystyle\int_1^3f(x)\,\mathrm dx\approx\frac12\sum_{i=1}^4f(r_i)=\frac12\left(\frac{27}4+5+\frac{11}4+0\right)=\boxed{\frac{29}4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_1%5E3f%28x%29%5C%2C%5Cmathrm%20dx%5Capprox%5Cfrac12%5Csum_%7Bi%3D1%7D%5E4f%28r_i%29%3D%5Cfrac12%5Cleft%28%5Cfrac%7B27%7D4%2B5%2B%5Cfrac%7B11%7D4%2B0%5Cright%29%3D%5Cboxed%7B%5Cfrac%7B29%7D4%7D)