- #1
Gotcha
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Can anyone please direct me in the right way on working out the approximate area of a semi-circle with equation y = (r^2 - x^2)^0.5, by using a Riemann Sum
How to Approximate the Area of a Semi-Circle Using Riemann Sums
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- Thread starter Gotcha
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SUMMARY
The discussion focuses on approximating the area of a semi-circle using Riemann sums, specifically the function y = (r^2 - x^2)^0.5. Participants emphasize the importance of selecting an appropriate domain, which is defined as [-3, 3], and creating an arbitrary partition of that domain with points labeled {x_0, x_1,...x_n}. The Riemann sum is constructed using the formula Σ_{i=1}^{n} y(t_i)(x_{i}-x_{i-1}), where t_i are points within the intervals [x_{i-1}, x_{i}]. The discussion also highlights the necessity of ensuring all partition values are positive to avoid potential issues.
PREREQUISITES- Understanding of Riemann sums and their application in definite integrals.
- Familiarity with the equation of a semi-circle, specifically y = (r^2 - x^2)^0.5.
- Basic knowledge of coordinate systems and domain selection.
- Ability to construct and evaluate mathematical sums.
- Research the concept of Riemann sums in greater detail, focusing on their application in calculus.
- Learn how to implement numerical integration techniques using software tools or programming languages.
- Explore the implications of partitioning in Riemann sums and how it affects approximation accuracy.
- Study the properties of semi-circles and their equations for further mathematical applications.
Students studying calculus, mathematics educators, and anyone interested in numerical methods for approximating areas under curves.
- #2
dextercioby
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What do you know about Riemann sums (meaning their general formula for the case of simple definite integrals)...?
Chose a system of coordinates with the center at the left end of the semicircle,so that the Ox axis in its posotive part to comprise entire diameter.Therefore your equation for the curve will be slightly modified.
Daniel.
Chose a system of coordinates with the center at the left end of the semicircle,so that the Ox axis in its posotive part to comprise entire diameter.Therefore your equation for the curve will be slightly modified.
Daniel.
- #3
quasar987
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Here's what I suggest..
First find the domain of that function ([-3,3]). Then create an arbitrary partition of that domain. That is to say, select an arbitrary number of points in that domain and label them \{x_0, x_1,...x_n\}. (x0 has to be -3 and xn has to be 3). In principle, the more points you chose, the better the approximation.
Then construct and evaluate the Riemann sum
\Sigma_{i=1}^{n} y(t_i)(x_{i}-x_{i-1})
Where the ti are arbitrarily chosen points in the interval [x_{i-1},x_{i}][/tex]
First find the domain of that function ([-3,3]). Then create an arbitrary partition of that domain. That is to say, select an arbitrary number of points in that domain and label them \{x_0, x_1,...x_n\}. (x0 has to be -3 and xn has to be 3). In principle, the more points you chose, the better the approximation.
Then construct and evaluate the Riemann sum
\Sigma_{i=1}^{n} y(t_i)(x_{i}-x_{i-1})
Where the ti are arbitrarily chosen points in the interval [x_{i-1},x_{i}][/tex]
- #4
dextercioby
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Yes,but i suggested him that all values of the partition be positive,the way you took'em half are and half are not...I think that should create some avoidable problems...
Daniel.
Daniel.
- #5
quasar987
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Really? Like what? I don't see what the difference will be, since x_{i}-x_{i-1} will be positive anyway.
- #6
Aki
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I got a cool program on my graphing calculator that does that for me. It's handy when it comes to test. If you want it, just pm me.
- #7
dextercioby
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I don't know who you were talking to,but if,by chance,u were talking to me,learn that i don't have a graphing computer and that's why the software would be totally useless...
Thanks for the offer,though...
Daniel.
Thanks for the offer,though...
Daniel.
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