1. 4.2/4.6 Approximating Area
Mt. Shasta, California

2. Consider an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance:
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
time
velocity
After 4 seconds, the object has gone 12 feet.

3. If the velocity is not constant,
we might guess that the
distance traveled is still equal
to the area under the curve.
(The units work out.)
Example:
We could estimate the area under the curve by drawing rectangles touching at their left corners.
This is called the Left-hand Reimann Sum(LHS).
Approximate area:

4. Left Hand Reimann Sum:
( h = width of subinterval )

5. We could also use a Right-hand Reimann Sum (RHS)
Approximate area:

6. Right Hand Reimann Sum:
( h = width of subinterval )

7. We could use the midpoint of each rectangle as the heights
We could use the midpoint of each rectangle as the heights. This is called the midpoint rule.
Averaging right and left rectangles gives us trapezoids:

9. Midpoint Rule: ( h = width of subinterval )
This sometimes gives us a better approximation than either left or right rectangles.

10. We could split the area into 4 trapezoids and add up the area of each trapezoid.
Averaging right and left rectangles gives us trapezoids:

12. Trapezoidal Rule: ( h = width of subinterval )
This can give us a better approximation than either left or right rectangles.