Integral of Bell Curve: Area of Slices

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SUMMARY

The integral of the function e^{-r^{2}} from negative to positive infinity represents the area under the bell curve, specifically the area of vertical slices through its center. This integral is commonly used in various applications, including calculating work, flux, and centroids. The discussion highlights a misunderstanding regarding the orientation of the slices, clarifying that the slices can be interpreted as passing through the origin rather than being parallel to the y-axis. The relationship between the area under the curve and the volume of the rotated bell curve about the z-axis is also emphasized.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with the Gaussian function e^{-x^{2}}
  • Knowledge of volume calculations using the method of slices and shells
  • Basic concepts of multivariable calculus
NEXT STEPS
  • Study the properties of the Gaussian integral and its applications
  • Learn about the method of cylindrical shells for volume calculations
  • Explore the concept of flux in vector calculus
  • Investigate the relationship between integrals and areas in multivariable functions
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Students and educators in mathematics, particularly those studying calculus and its applications in physics and engineering, will benefit from this discussion.

icystrike
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Can i ask what is the area we are referring to when we take such integral (r is the radius):

\int_{-\infty }^{\infty }e^{-r^{2}}dr

I'm suspecting that its is the area of slices of bell curve that rotates about the z-axis.

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hi icystrike! :smile:

it's the integral of a vertical slice of the bell curve through its centre :wink:
 
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!
 
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 - A^{2}=\pi )
 
hi icystrike! :smile:
icystrike said:
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

i think he means that it'll be the same (it's the same shape), apart from a factor e-x2 :wink:
 
Oh! Thats what he meant! Truly enlighten! Thanks Tim! :)
(Came to ensure that i get the concept right)
 

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