CALCULUS: AREA UNDER A CURVE

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CALCULUSAREA UNDER A CURVE
Final Project C I 336 Terry Kent
The calculus is the greatest aid we have to the
application of physical truth. W.F. Osgood
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RULE OF 4

• VERBALLY
• GRAPHICALLY (VISUALLY)
• NUMERICALLY
• SYMBOLICLY (ALGEBRAIC CALCULUS)

Calculus is the most powerful weapon of thought
yet devised by the wit of man. W.B. Smith
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VERBAL PROBLEM

• Find the area under a curve bounded by the curve,
  the x-axis, and a vertical line.
• EXAMPLE Find the area of the region bounded by
  the curve y x2, the x-axis, and the line x 1.

Do or do not. There is no try. -- Yoda
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GRAPHICALLY
Mathematics consists of proving the most obvious
thing in the least obvious way George Polya
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NUMERICALLY
The area can be approximated by dividing the
region into rectangles. Why rectangles? Easiest
area formula! Would there be a better figure to
use? Trapezoids! Why not use them?? Formula too
complex !!
The essence of mathematics is not to make simple
things complicated, but to make complicated
things simple. -- Gudder
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AREA BY RECTANGLES
Exploring Riemann Sums Approximate the area using
5 rectangles.
Left-Hand Area .24
Right-Hand Area .444
Midpoint Area .33
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NUMERICALLY
AREA IS APPROACHING 1/3 !!
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ADDITIONAL EXAMPLES

• Approximate the area under the curve using 8
  left-hand rectangles for f(x) 4x - x2, 0,4.
• A 10.75
• Approximate the area under the curve using 6
  right-hand rectangles for f(x) x3 2, 0,2.
• A 9.444
• Approximate the area under the curve using 10
  midpoint rectangles for f(x) x3 - 3x2 2,
  0,4.
• A 7.84

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SYMBOLICLYALGEBRAIC
How could we make the approximation more exact?
More rectangles!! How many rectangles would we
need? ???
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SYMBOLICLYALGEBRAIC
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ADDITIONAL EXAMPLES

• Use the Limit of the Sum Method to find the area
  of the following regions
• f(x) 4x - x2, 0,4. A 32/3
• f(x) x3 2, 0,2. A 8
• f(x) x3 - 3x2 2, 0,4. A 8

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SYMBOLICALYCALCULUS
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CONCLUSION
The Area under a curve defined as y f(x) from
x a to x b is defined to be
Thus mathematics may be defined as the subject
in which we never know what we are talking about,
not whether what we are saying is true. --
Russell
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ADDITIONAL EXAMPLES

• Use Integration to find the area of the
  following regions
• f(x) 4x - x2, 0,4. A 32/3
• f(x) x3 2, 0,2. A 8
• f(x) x3 - 3x2 2, 0,4. A 8

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AREA APPLICATION
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FUTURE TOPICS

• PROPERTIES OF DEFINITE INTEGRALS
• AREA BETWEEN TWO CURVES
• OTHER INTEGRAL APPLICATIONS
• VOLUME, WORK, ARC LENGTH
• OTHER NUMERICAL APPROXIMATIONS
• TRAPEZOIDS, PARABOLAS

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REFERENCES

• CALCULUS, Swokowski, Olinick, and Pence, PWS
  Publishing, Boston, 1994.
• MATHEMATICS for Everyman, Laurie Buxton, J.M.
  Dent Sons, London, 1984.
• Teachers Guide AP Calculus, Dan Kennedy, The
  College Board, New York, 1997.
• www.archive,math.utk.edu/visual.calculus/
• www.cs.jsu.edu/mcis/faculty/leathrum/Mathlet/riema
  nn.html
• www.csun.edu/hcmth014/comicfiles/allcomics.html

People who dont count, dont count. -- Anatole
France