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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
HOME WORKArea Approximations worksheet