Area under Curves

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Area under Curves

• 5.2

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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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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 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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Inscribed rectangles are all below the curve
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We will be learning how to find the exact area
under a curve if we have the equation for the
curve. Rectangular approximation methods are
still useful for finding the area under a curve
if we do not have the equation.
Area ((b-a)/2)f(x0)f(x1)f(xn)
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Trapezoidal Rule
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What if
We could split the area under the curve into a
lot of thin trapezoids.
It seems reasonable that the distance will equal
the area under the curve.
Area .5(b1b2)h
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Example

• Find the area under x3 using 4 subintervals
  using left, right, midpoint and trapezoidal
  methods from 2, 3

2
3
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Example

• Find the area under x3 using 4 subintervals
  using left, right, midpoint and trapezoidal
  methods from 2, 3

2
3
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Example

• Find the area under x3 using 4 subintervals
  using left, right, midpoint and trapezoidal
  methods from 2, 3

2
3
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Example

• Find the area under x3 using 4 subintervals
  using left, right, midpoint and trapezoidal
  methods from 2, 3

2
3
Actual Area 16.25