area approximation
DESCRIPTION
Area Approximation. 4-B. Exact Area. Use geometric shapes such as rectangles, circles, trapezoids, triangles etc…. rectangle. parallelogram. triangle. Midpoint Trapezoidal Rule. Approximate Area. Riemann sums Left endpoint Right endpoint . Approximate Area. - PowerPoint PPT PresentationTRANSCRIPT
AREA APPROXIMATION
4-B
Exact Area
Use geometric shapes such as rectangles, circles, trapezoids, triangles etc…
rectangle
triangle
parallelogram
Approximate Area
• Midpoint
• Trapezoidal Rule•
)(21
21 bbhAtrap
)2...22(21
1210 nnT yyyyynabA
)...(2
122
52
32
1
nM yyyynabA
Approximate Area
• Riemann sums• Left endpoint
• Right endpoint
)...( 1210
nLE yyyynabA
)...( 321 nRE yyyynabA
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
nabdx
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
bpbapapabA
CBxAxxp
)(4...4)(24(3 13210 nn xfxfxfxfxfxfnabA
1) Find the area under the curve from
229)( xxf
12 x2rA
2) Approximate the area under fromWith 4 subintervals using inscribed rectangles
2sin)( xxf
23
2
x
)...( 321 nRE yyyynabA
2
43
45
23
3) Approximate the area under fromUsing the midpoint formula and n = 4
24 xy
11 x
43
21
41
0 11 41
21 4
3
)...(2
122
52
32
1
nM yyyynabA
4) Approximate the area under the curve between x = 0 and x = 2Using the Trapezoidal Rule with 6 subintervals
26 xy
31
1034 2
32
35
)2...22(21
1210 nnT yyyyynabA
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 xfxfxfxfxfxfnabA
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
123
245
21 x
470
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