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CALCULUS: AREA UNDER A CURVE

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... the area under the curve using 8 left-hand rectangles for f(x) = 4x - x2, ... How many rectangles would we need? ??? SYMBOLICLY: ALGEBRAIC. ADDITIONAL EXAMPLES ... – PowerPoint PPT presentation

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Title: CALCULUS: AREA UNDER A CURVE


1
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
2
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
3
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
4
GRAPHICALLY
Mathematics consists of proving the most obvious
thing in the least obvious way George Polya
5
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
6
AREA BY RECTANGLES
Exploring Riemann Sums Approximate the area using
5 rectangles.
Left-Hand Area .24
Right-Hand Area .444
Midpoint Area .33
7
NUMERICALLY
AREA IS APPROACHING 1/3 !!
8
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

9
SYMBOLICLYALGEBRAIC
How could we make the approximation more exact?
More rectangles!! How many rectangles would we
need? ???
10
SYMBOLICLYALGEBRAIC
11
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

12
SYMBOLICALYCALCULUS
13
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
14
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

15
AREA APPLICATION
16
FUTURE TOPICS
  • PROPERTIES OF DEFINITE INTEGRALS
  • AREA BETWEEN TWO CURVES
  • OTHER INTEGRAL APPLICATIONS
  • VOLUME, WORK, ARC LENGTH
  • OTHER NUMERICAL APPROXIMATIONS
  • TRAPEZOIDS, PARABOLAS

17
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
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