6.3 - Approximating Areas
I. Suppose you were driving a car at a constant rate of 55 mph for 2 hours. Since d  v  t , the total
distance travelled would be d = (55)(2) = 110 mph.
II. What if velocity (rate) is not constant?
Example: Approximate the area between f ( x)  x 2  1 and the x-axis on [1, 4] using 3 rectangles.
x = width of each rectangle
Notation: n = # of rectangles
f ( xi ) = height of the i th
rectangle
Approximate Area 
n
 f (x ) x .
i 1
Graph:
i
A. Left sum uses left endpoint of rectangle as height.
B. Right sum uses right endpoint of the rectangle as height.
C. Midpoint uses the midpoint of the rectangle as height.
This is an approximation of the signed area between a function and the x-axis on a given interval that is
found using signed areas of rectangles.
Hence, left sums, right sums and midpoint sums are special cases of Riemann sums.