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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? ???

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
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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