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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)

“Calculus is the most powerful weapon of thought yet devised by the wit of man.” – W.B. Smith

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

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

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

Exploring Riemann Sums

Approximate the area using 5 rectangles.

Left-Hand Area = .24

Right-Hand Area = .444

Midpoint Area = .33

Left EndpointInscribed Rectangles

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

Right EndpointCircumscribed Rectangles

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

Midpoint

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

NUMERICALLY

AREA IS APPROACHING 1/3 !!

ADDITIONAL EXAMPLES

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

A =

ADDITIONAL EXAMPLES

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

A =

ADDITIONAL EXAMPLES

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

A =

SYMBOLICLY:ALGEBRAIC

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

How many rectangles would we need? ???

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

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 EXAMPLES

Use Integration to find the area of the following regions:

- f(x) = 4x - x2, [0,4].

A =

ADDITIONAL EXAMPLES

Use Integration to find the area of the following regions:

- f(x) = x3 + 2, [0,2].

A =

ADDITIONAL EXAMPLES

Use Integration to find the area of the following regions:

- f(x) = x3 - 3x2 + 2, [0,4].

A =

FUTURE TOPICS

PROPERTIES OF DEFINITE INTEGRALS

AREA BETWEEN TWO CURVES

OTHER INTEGRAL APPLICATIONS:

VOLUME, WORK, ARC LENGTH

OTHER NUMERICAL APPROXIMATIONS:

TRAPEZOIDS, PARABOLAS

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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