1. 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

2. RULE OF 4 VERBALLY GRAPHICALLY GRAPHICALLY (VISUALLY) NUMERICALLY SYMBOLICLYSYMBOLICLY (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 = x 2, 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. Left Endpoint Inscribed Rectangles n=# rectangles a= left endpoint b=right endpoint

8. Right Endpoint Circumscribed Rectangles n=# rectangles a= left endpoint b=right endpoint

9. Midpoint n=# rectangles a= left endpoint b=right endpoint

10. NUMERICALLY AREA IS APPROACHING 1/3 !!

11. ADDITIONAL EXAMPLES Approximate the area under the curve using 8 left- hand rectangles for f(x) = 4x - x 2, [0,4]. A =

12. ADDITIONAL EXAMPLES Approximate the area under the curve using 6 right-hand rectangles for f(x) = x 3 + 2, [0,2]. A =

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

14. SYMBOLICLY: ALGEBRAIC How could we make the approximation more exact? More rectangles!! How many rectangles would we need? ???

15. SYMBOLICLY: ALGEBRAIC

16. ADDITIONAL EXAMPLES Use the Limit of the Sum Method to find the area of the following regions: f(x) = 4x - x 2, [0,4].A = 32/3 f(x) = x 3 + 2, [0,2].A = 8 f(x) = x 3 - 3x 2 + 2, [0,4].A = 8

17. SYMBOLICALY: CALCULUS

18. 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

19. ADDITIONAL EXAMPLES Use Integration to find the area of the following regions: f(x) = 4x - x 2, [0,4]. A =

20. ADDITIONAL EXAMPLES Use Integration to find the area of the following regions: f(x) = x 3 + 2, [0,2]. A =

21. ADDITIONAL EXAMPLES Use Integration to find the area of the following regions: f(x) = x 3 - 3x 2 + 2, [0,4]. A =

22. AREA APPLICATION

23. FUTURE TOPICS PROPERTIES OF DEFINITE INTEGRALS AREA BETWEEN TWO CURVES OTHER INTEGRAL APPLICATIONS: VOLUME, WORK, ARC LENGTH OTHER NUMERICAL APPROXIMATIONS: TRAPEZOIDS, PARABOLAS

24. 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