The average value of [tex]f(x)[/tex] over the given interval is
[tex]\dfrac1{20-10}\displaystyle\int_{10}^{20}f(x)\,\mathrm dx[/tex]
You're given five points, but only four will contribute to the sum as there are four subintervals that you can work with. Because you are finding the right-endpoint approximation, the height of the rectangles used in the Riemann sum will be determined by the points [tex]x\in\{12,15,19,20\}[/tex], where the height of each rectangle is the corresponding value of [tex]f(x)[/tex] and the width is the length of the subinterval. In this case, the lengths would be (left to right and respectively) [tex]\Delta x\in\{2,3,4,1\}[/tex], which is simply the set of forward differences of the previous set.
So the average value is approximated by
[tex]\dfrac1{20-10}\displaystyle\sum_{x\in\{12,15,19,20\}}f(x)\Delta x=\frac1{10}\left(2f(12)+3f(15)+4f(19)+f(20)\right)=-\dfrac{101}{10}=-10.1[/tex]
