Hello Tilda,
For the by rectangles, we have to divide the section [0,2] into 8 subdivisions and total their areas. The width of the rectangles is (2-0)/8=1/4. So we need to find all values of this function from 0 through 2 on intervals of 1/4.
f(x)=√x+2
f(0)=2
f(1/4)=5/2
f(1/2)=2+√2/2
f(3/4)=2+√3/2
f(1)=3
f(5/4)=2+√5/2
f(3/2)=2+√6/2
f(7/4)=2+7(√7/2)
f(2)=2+√2
These values are the heights of each rectangle. The width of each rectangle is 1/4. Now for the lower sum, we take the area of the all rectangles from the left side 1/4(f(0)) + 1/4(f(1/4) +1/4(f(1/2) ... +1/4(f(7/4). Note that I stopped at the the 7/4 rather than two because I only want the 8 rectangles from the left side.
For the upper sum, I start at the second value and do the same. 1/4(f(1/4) +1/4(f(1/2) + 1/4f(3/4) ... +1/4(f(2).
I'm not actually going to demonstrate this to completion here because it is a little tedious, and this is your homework after all, but it does help if you factor out the 1/4 to make it 1/4(f(0) + f(1/4) + f(1/2)... f(7/4).
