Question

can anyone help me to differentiate between lower(upper) sum and lower(upper) integral..

Answer 1

Let's look at an example
Say you have the function y=x^2 and you want to find area under that curve from 0 to 2.
And it is asked you find the approximate area using an upper sum and then the lower sum.
Well let me draw both of those pictures:
Say we are to do this with 2 rectangles.

So (2-0)/2=1=length of base of each rectangle



I put lower sum pic in that one picture because all of the rectangles occur below the curve so when I add up the areas of those rectangles I will get a lower approximate sum of the area.

And in the upper sum pic, all the rectangles occur above the curve which means I will get an upper approximate sum of the area.

And of course the real area will occur between the lower and the upper approximate areas.