A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles,...
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Transcript of A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles,...
![Page 1: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/1.jpg)
AREA APPROXIMATION
4-E Riemann Sums
![Page 2: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/2.jpg)
Exact Area
Use geometric shapes such as rectangles, circles, trapezoids, triangles etc…
rectangletriangle
parallelogramcircle
![Page 3: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/3.jpg)
Approximate Area: Add the area of the Rectangles
• Midpoint
)...(2
122
52
32
1
nM hhhhn
abA
x
y
![Page 4: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/4.jpg)
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
![Page 5: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/5.jpg)
Approximate Area
• Riemann sums• Left endpoint
• Right endpoint
)...( 1210
nLE hhhhn
abA
)...( 321 nRE hhhhn
abA
![Page 6: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/6.jpg)
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
![Page 7: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/7.jpg)
Left endpoints:Increasing: inscribedDecreasing: circumscribed
Right Endpoints: increasing: circumscribed, decreasing: inscribed
)...( 321 nRE hhhhn
abA
![Page 8: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/8.jpg)
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
![Page 9: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/9.jpg)
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
![Page 10: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/10.jpg)
1) Find the area under the curve from 32 x
![Page 11: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/11.jpg)
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
![Page 12: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/12.jpg)
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
![Page 13: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/13.jpg)
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
![Page 14: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/14.jpg)
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
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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
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7) Approximate by using 5 rectangles of equal width and an Upper Riemann Sum
10
0
)( dxxf
0
x
y
![Page 17: A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.](https://reader035.fdocuments.in/reader035/viewer/2022062519/5697bfc41a28abf838ca6223/html5/thumbnails/17.jpg)
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