AREA APPROXIMATION

4-E Riemann Sums

Exact Area

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

rectangletriangle

parallelogramcircle

Approximate Area: Add the area of the Rectangles

• Midpoint

)...(2

122

52

32

1

nM hhhhn

abA

x

y

Approximate Area: Add the area of the Rectangles

Trapezoidal Rule

)(2

121 bbwAtrap

)2...22(2

11210 nnT hhhhh

n

abA

0 1 2 11

2( ) ( 2 2 ... 2 )

b

n naf x dx w h h h h h

x

y

5432

1

0x 1x 2x 1nx nx

0h

y f x

1h

2h nh1nh

a b

x

y

Approximate Area

• Riemann sums• Left endpoint

• Right endpoint

)...( 1210

nLE hhhhn

abA

)...( 321 nRE hhhhn

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

)...( 1210

nLE hhhhn

abA

Left endpoints:Increasing: inscribedDecreasing: circumscribed

Right Endpoints: increasing: circumscribed, decreasing: inscribed

)...( 321 nRE hhhhn

abA

The exact 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

1) Find the area under the curve from 32 x

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

2sin)( xxf

2

3

2

x

2

4

3 4

52

3

)...( 321 nRE hhhhn

abA

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 hhhhn

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 hhhhh

n

abA

5) 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

6) Find approximate the area under the curve on the interval using right hand Riemann sum with 4 equal subdivisions

22 xxy 21 x

12

32

4

5

4

7

0

7) Approximate by using 5 rectangles of equal width and an Upper Riemann Sum

10

0

)( dxxf

0

x

y

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