AREA APPROXIMATION

4-B

Exact Area

Use geometric shapes such as rectangles, circles, trapezoids, triangles etc…

rectangle

triangle

parallelogram

Approximate Area

• Midpoint

• Trapezoidal Rule•

)(2

121 bbhAtrap

)2...22(2

11210 nnT yyyyy

n

abA

)...(2

122

52

32

1

nM yyyyn

abA

Approximate Area

• Riemann sums• Left endpoint

• Right endpoint

)...( 1210

nLE yyyyn

abA

)...( 321 nRE yyyyn

abA

Inscribed Rectangles: rectangles remain under the curve. Slightly underestimates the area.

Circumscribed Rectangles: rectangles are slightly above the curve. Slightly overestimates the area Left Endpoints

Left endpoints:Increasing: inscribedDecreasing: circumscribed

Right Endpoints: increasing: circumscribed, decreasing: inscribed

The area under a curve bounded by f(x) and the x-axis and the linesx = a and x = b is given by

Where

and n is the number of sub-intervals

n

i

dxxfn 1

)(lim

n

abdx

Therefore:

n

i

n

i

dxxfregionofareadxxf1

21

1)(

Inscribed rectangles

Circumscribed rectangles

http://archives.math.utk.edu/visual.calculus/4/areas.2/index.html

The sum of the area of the inscribed rectangles is called a lower sum, and the sum of the area of the circumscribed rectangles is called an upper sum

Fundamental Theorem of Calculus:If f(x) is continuous at every point [a, b] and F(x) is an antiderivative of f(x) on [a, b] then the area under the curve can be approximated to be

b

a

aFbFdxxf )()()(

n

i

dxxfn 1

)(lim

-

+

Simpson’s Rule:

)(2

4)(6

then )( if 2

bpba

papab

A

CBxAxxp

)(4...4)(24(3 13210 nn xfxfxfxfxfxfn

abA

1) Find the area under the curve from

229)( xxf

12 x

2rA

2) Approximate the area under fromWith 4 subintervals using inscribed rectangles

2sin)( xxf

2

3

2

x

)...( 321 nRE yyyyn

abA

2

4

3 4

52

3

3) Approximate the area under fromUsing the midpoint formula and n = 4

24 xy

11 x

4

3

2

1 4

1 0 11 4

1

2

1 4

3

)...(2

122

52

32

1

nM yyyyn

abA

4) Approximate the area under the curve between x = 0 and x = 2Using the Trapezoidal Rule with 6 subintervals

26 xy

3

110

3

4 23

2

3

5

)2...22(2

11210 nnT yyyyy

n

abA

5) Use Simpson’s Rule to approximate the area under the curve on the interval using 8 subintervals

3)( xxf

1 30 4 62 5

80 x

)(4...4)(24(3 13210 nn xfxfxfxfxfxfn

abA

7 8

6) The rectangles used to estimate the area under the curve on the interval

using 5 subintervals with right endpoints will bea) Inscribedb) Circumscribedc) Neitherd) both

3)( xxf 83 x

7) Find the area under the curve on the interval using 4 inscribed rectangles

22 xxy

12

32

4

5

21 x

4

70

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