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CALCULUS: AREA 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. RULE OF 4 . VERBALLY GRAPHICALLY (VISUALLY) NUMERICALLY SYMBOLICLY (ALGEBRAIC & CALCULUS).

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Calculus area under a curve
CALCULUS:AREA 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


Rule of 4
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


Verbal problem
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


Graphically
GRAPHICALLY

“Mathematics consists of proving the most obvious thing in the least obvious way” – George Polya


Numerically
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


Area by rectangles
AREA BY RECTANGLES

Exploring Riemann Sums

Approximate the area using 5 rectangles.

Left-Hand Area = .24

Right-Hand Area = .444

Midpoint Area = .33


Left endpoint inscribed rectangles
Left EndpointInscribed Rectangles

n=# rectangles a= left endpoint b=right endpoint


Right endpoint circumscribed rectangles
Right EndpointCircumscribed Rectangles

n=# rectangles a= left endpoint b=right endpoint


Midpoint
Midpoint

n=# rectangles a= left endpoint b=right endpoint


Numerically1
NUMERICALLY

AREA IS APPROACHING 1/3 !!


Additional examples
ADDITIONAL EXAMPLES

  • Approximate the area under the curve using 8 left-hand rectangles for f(x) = 4x - x2, [0,4].

    A =


Additional examples1
ADDITIONAL EXAMPLES

  • Approximate the area under the curve using 6 right-hand rectangles for f(x) = x3 + 2, [0,2].

    A =


Additional examples2
ADDITIONAL EXAMPLES

  • Approximate the area under the curve using 10 midpoint rectangles for f(x) = x3 - 3x2 + 2, [0,4].

    A =


Symbolicly algebraic
SYMBOLICLY:ALGEBRAIC

How could we make the approximation more exact? More rectangles!!

How many rectangles would we need? ???


Symbolicly algebraic1
SYMBOLICLY:ALGEBRAIC


Additional examples3
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


Symbolicaly calculus
SYMBOLICALY:CALCULUS


Conclusion
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


Additional examples4
ADDITIONAL EXAMPLES

Use Integration to find the area of the following regions:

  • f(x) = 4x - x2, [0,4].

    A =


Additional examples5
ADDITIONAL EXAMPLES

Use Integration to find the area of the following regions:

  • f(x) = x3 + 2, [0,2].

    A =


Additional examples6
ADDITIONAL EXAMPLES

Use Integration to find the area of the following regions:

  • f(x) = x3 - 3x2 + 2, [0,4].

    A =



Future topics
FUTURE TOPICS

PROPERTIES OF DEFINITE INTEGRALS

AREA BETWEEN TWO CURVES

OTHER INTEGRAL APPLICATIONS:

VOLUME, WORK, ARC LENGTH

OTHER NUMERICAL APPROXIMATIONS:

TRAPEZOIDS, PARABOLAS


References
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/riemann.html

  • www.csun.edu/~hcmth014/comicfiles/allcomics.html

“People who don’t count, don’t count.” -- Anatole France


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