The 4 subintervals are [6,7], [7,8], [8,9], [9,10].
Instead of using calculus we can use the left endpoint as the height of a rectangle of unit width.
So the area is f(6)+f(7)+f(8)+f(9)=43.92+36.77+36.92+47.97=165.58.
(Using calculus the actual integral gives 176.896)