Problem :
Find
f (x_{k})Δx for f (x) = 2x on [0, 2] 

To solve this problem, notice that the graph of
f (x) is a line, and the
area in question is in the shape of a right triangle with base 2 and
height
f (2) = 4. So, the limit of the right hand sum, which
is the area under the curve, is
(2)(4) = 4 

Problem :
Find
f (x_{k})Δx for f (x) = on [0, 3] 

To solve this problem, notice that the graph of
f (x) is a semicircle of
radius 3 centered at the origin. The interval
[0, 3] contains a section
of the curve that is equivalent to a
quartercircle. Thus, the area under the curve is equal to onefourth the
area of a circle
with radius 3, or
Π(3^{2}) = Π.