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CALCULUS: AREA UNDER A CURVEPowerPoint Presentation

CALCULUS: AREA UNDER A CURVE

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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? ???

SYMBOLICLY:ALGEBRAIC

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

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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