Section 4.2 Area and Estimating with Finite Sums

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Section 4.2Area and Estimating with Finite
Sums
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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.
After 4 seconds, the object has gone 12 feet.
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Consider this same object moving at a varying
rate defined by the function f(x) x
Distance is still found using rate . time
distance
That means the area under the curve STILL
represents the distance traveled, but since we
dont have a constant rate, how could we find
that area now (after 4 seconds)?
Heck its a triangle! Thats easy!
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What happens, though, if the velocity function
is not constant, nor does it form a regular
geometric shape (with an easy area formula)
when we shade the area underneath it? Whats a
student to do?? How can we find the area under
that kind of curve??
Example
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Well finding the area of that first rectangle
was pretty easy. Why dont we use rectangles to
try and approximate the area of this unusual
shape?
What a brilliant idea! ?
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If the velocity is not constant, we might guess
that the distance traveled is still equal to the
area under the curve.
(The units do 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 Rectangular
Approximation Method (LRAM).
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We could also use a Right-hand Rectangular
Approximation Method (RRAM).
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Another approach would be to use rectangles that
touch at the midpoint. This is the Midpoint
Rectangular Approximation Method (MRAM).
In this example there are four subintervals. As
the number of subintervals increases, so does the
accuracy.
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With 8 subintervals
width of subinterval
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• Using Rectangular Approximation Methods in this
  example
• LRAM produced smaller than true area
• RRAM produced larger than true area
• MRAM produced closest to true area

More Accurate More intervals
MRAM10 - means 10 slices or subintervals
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• Rectangular Approximation Methods
• LRAM -
• RRAM -
• MRAM -

RAM is used to find the area under the curve.
Left RAM Right RAM Midpoint RAM
The natural question to follow might be, Which
one should I use? It seems like the MRAM gave
the best estimate, so should I always use that
one?
Well lets see
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Lets use f(x) x2sinx as an example and use RAM
to estimate the area under the curve from x 0
to x 3.
n LRAM MRAM RRAM
5 5.15480 5.89668 5.91685
10 5.52574 5.80685 5.90677
25 5.69079 5.78150 5.84320
50 5.73615 5.77788 5.81235
100 5.75701 5.77697 5.79511
1,000 5.77476 5.77667 5.77857
As n gets increasingly larger, ALL of the
approximations approach the same value!
The actual area is 5.77666752.
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• A particle starts at x0 and moves along the
  x-axis with velocity v(t)t2 for time t gt0.
  Where is the particle at t3? (use 6 intervals)

LRAM y x y x y x ... v t v t v t ...
02(1/2) (1/2)2(1/2) (1)2(1/2) (1½)2(1/2)
(2)2(1/2) (21/2)2(1/2) 6.875
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• A particle starts at x0 and moves along the
  x-axis with velocity v(t)t2 for time t gt0.
  Where is the particle at t3? (use 6 intervals)

RRAM y x y x y x ... v t v t v t ...
(1/2)2(1/2) (1)2(1/2) (1½)2(1/2)
(2)2(1/2) (21/2)2(1/2) (3)2(1/2) 11.375
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• A particle starts at x0 and moves along the
  x-axis with velocity v(t)t2 for time t gt0.
  Where is the particle at t3? (use 6 intervals)

MRAM y x y x y x ... v t v t v t ...
(1/4)2(1/2) (3/4)2(1/2) (11/4)2(1/2)
(13/4)2(1/2) (21/4)2(1/2) (23/4)2(1/2)
8.9375
The actual area is 9.
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Area Under A Curve

• The area under a curve on an interval can be
  approximated by summing the areas of individual
  rectangles on the interval.

• If the rectangles are inscribed under the curve,
  the approximated area will be less than the
  desired area and the method will provide an
  underestimation.

• If the rectangles are circumscribed above the
  curve, the area will be greater than the desired
  area and the method will provide an
  overestimation.

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Area Under A Curve

• By using a finer partition (one with more
  subintervals), we make each rectangle narrower
  (increasing their number), and we get an area
  value that is closer to the true area under the
  curve.
• By taking the limit as n (the number of
  rectangles) approaches infinity, the actual area
  is approached.
• Lets look

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Let f(x) be a nonnegative, continuous function on
the closed interval a, b. Then, the area of
the region under the graph of f(x) is given
by
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ASSIGNMENTSection 4.2 Worksheet(1 6, 15,
17)