CALCULUS:AREA UNDER A CURVE

Final ProjectC & I 336

Terry Kent

“The calculus is the greatest aid we have to the application of physical truth.” – W.F. Osgood

RULE OF 4

VERBALLYGRAPHICALLY (VISUALLY)NUMERICALLYSYMBOLICLY (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 RECTANGLESExploring Riemann SumsApproximate 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 b ax

n

Right EndpointCircumscribed Rectangles

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

n

Midpoint

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

n

NUMERICALLYSum of areas of Sum of areas of

inscribed rectangles

circumscribed rectangles

2 0.125 0.625 0.3754 0.21875 0.46875 0.343758 0.2734375 0.3984375 0.335937516 0.30273438 0.36523438 0.3339843832 0.31787109 0.34912109 0.3334960964 0.32556152 0.34118652 0.33337402128 0.32943726 0.33724976 0.33334351256 0.33138275 0.335289 0.33333588512 0.33235741 0.33431053 0.333333971024 0.33284521 0.33382177 0.33333349

nAverage of the

Two Sums

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

( ) , where ib a

A f x x xn

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

How many rectangles would we need? ???

00

lim ( ) , where n

nix

b aA f a i x x x

n

SYMBOLICLY:ALGEBRAIC

00

2

0

2

30

3

lim ( ) , where

1 1lim 0

lim

1 ( 1)(2 1) 1lim

6 3

n

nix

n

ni

n

ni

n

b aA f a i x x x

n

in n

i

n

n n n

n

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

12

0

3 1

3 3

A x dx

x

CONCLUSIONThe Area under a curve defined as y = f(x) from

x = a to x = b is defined to be:

0

1

lim ( )

bn

xin a

A f a i x x f x dx

“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

( ) ( ) ( ) ( )

bb

a

a

A f x dx F x F b F a

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 =

AREA APPLICATION

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