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Method to Integrate sqrt(1-x^2)


Date: 9/11/96 at 16:21:20
From: Anonymous
Subject: Method to Integrate sqrt(1-x^2)

What is the integral of [sqrt(1-x^2)]dx from [-1,1]?


Date: 9/12/96 at 0:21:33
From: Doctor Pete
Subject: Re: Method to Integrate sqrt(1-x^2)

Hint:  Use a trigonometric substitution of the form x = f(u), dx = 
f'(u) du. Also, you might want to remember the identity


     (Cos[x]^2) = (1+Cos[2x])/2,

which comes from the half-angle formula

     Cos[x/2] = Sqrt[(1+Cos[x])/2] .

This method will take you through two substitutions, the second of 
which is obvious.  Apply the limits after you find the indefinite 
integral in terms of your original variable x, or alternatively, you 
can apply the substitutions to the limits as you go.  If you need more 
help, feel free to ask.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 9/13/96 at 14:5:18
From: Doctor Ceeks
Subject: Re: Method to Integrate sqrt(1-x^2)

Hi,

Another way is to recognize this as the area of the semi-circle of
unit radius...so that the answer is pi/2.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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