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 [tex]r_i[/tex] are {3/2, 2, 5/2, 3}, which we can index by the sequence
[tex]r_i=1+\dfrac i2[/tex]
with [tex]1\le i\le4[/tex].
Evaluating the function at the right endpoints gives the sampling points [tex]f(r_i)[/tex], {27/4, 5, 11/4, 0}.
Then the area is approximated by
[tex]\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}[/tex]
