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Integral of Bell Curve: Area of Slices
- Thread starter icystrike
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In summary, the conversation discusses the use of integrals to find the area of a rotated bell curve. The area is represented by the integral \int_{-\infty }^{\infty }e^{-r^{2}}dr, which is the integral of a vertical slice of the bell curve through its center. The conversation also mentions that integrals can be used to calculate other quantities like work, flux, and centroids. There is some confusion about whether the slice should be parallel to the y-axis or passing through the origin, but this is clarified as being the same shape with a factor e-x2.
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tiny-tim
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hi icystrike! 
it's the integral of a vertical slice of the bell curve through its centre
it's the integral of a vertical slice of the bell curve through its centre
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HallsofIvy
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You understand, I hope, that finding area is one possible application of the integral. When we calculate an integral we are not necessarily finding any area at all!
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icystrike
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Thanks tiny-tim and HallsofIvy!
Yes! I know that! We can use Integral to compute things like work, flux, centroids .. =D
Its just that my teacher actually relate the slice as "some slice that is parallel to the y-axis" while i think that it should be the slice that is passing through origin(He've probably made some mistake)... (My teacher was actually comparing the volume of a rotated bell curve about z axis by slice and shells to evaluate the area under bell curve - [tex]A^{2}=\pi[/tex] )
Yes! I know that! We can use Integral to compute things like work, flux, centroids .. =D
Its just that my teacher actually relate the slice as "some slice that is parallel to the y-axis" while i think that it should be the slice that is passing through origin(He've probably made some mistake)... (My teacher was actually comparing the volume of a rotated bell curve about z axis by slice and shells to evaluate the area under bell curve - [tex]A^{2}=\pi[/tex] )
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tiny-tim
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hi icystrike!
i think he means that it'll be the same (it's the same shape), apart from a factor e-x2
icystrike said:
i think he means that it'll be the same (it's the same shape), apart from a factor e-x2
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icystrike
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Oh! Thats what he meant! Truly enlighten! Thanks Tim! :)
(Came to ensure that i get the concept right)
(Came to ensure that i get the concept right)
1. What is the Bell Curve and why is it important in statistics?
The Bell Curve, also known as the Normal Distribution, is a symmetrical probability distribution that is commonly used in statistics. It is important because it can be used to model many natural phenomena, such as human height and IQ, and it allows for the calculation of probabilities and confidence intervals.
2. What is the Integral of a Bell Curve?
The Integral of a Bell Curve is the area under the curve, representing the total probability of all possible outcomes. It is calculated by finding the upper and lower limits of the integral and using integration techniques to solve the equation.
3. How is the Integral of a Bell Curve related to the Area of Slices?
The Integral of a Bell Curve is equal to the area of the slices, which are small sections of the curve that can be calculated using the integration technique. These slices represent the probability of a specific outcome occurring within a certain range of values.
4. How is the Integral of a Bell Curve used in real-world applications?
The Integral of a Bell Curve is used in various real-world applications, such as in finance to calculate stock market risks, in psychology to measure personality traits, and in medicine to predict the effectiveness of a treatment. It is also used in quality control to determine if a process is functioning within acceptable limits.
5. Are there any limitations to using the Integral of a Bell Curve?
While the Bell Curve is a useful tool, it is not always the best model for all data sets. It assumes that the data is normally distributed, which may not always be the case. Additionally, the Integral of a Bell Curve may not accurately represent the outliers in a data set, as it places more weight on the central values.