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 November 27th, 2008, 05:55 PM #1 Newbie   Joined: Nov 2008 Posts: 7 Thanks: 0 Determining bounds when changing variables Hello everybody! I have this question. Say I want to integrate sqrt(x^2-1) between 0 and sqrt(3)/2. I can substitute x=cos(x) to integrate -sin^2(x) between cos^(-1)(0) and cos^(-1)(sqrt(3)/2). But how do I choose which values to use? The integral of sin^2(x) is (x - sinx cosx)/2. Obviously taking cos^(-1)(0) = 3pi/2 or cos^(-1)(0) = pi/2 will give different answers. Now obviously cos^(-1)(x) is typically understood to give an answer in the range [0, pi]. But why wouldn't the integration work with the bound 3pi/2? I am a bit puzzled. Thanks a lot! Isaac P.S. Why not implement LaTeX with this forum?
November 27th, 2008, 06:12 PM   #2
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Re: Determining bounds when changing variables

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 Originally Posted by isaace Hello everybody! I have this question. Say I want to integrate sqrt(x^2-1) between 0 and sqrt(3)/2. I can substitute x=cos(x) to integrate -sin^2(x) between cos^(-1)(0) and cos^(-1)(sqrt(3)/2). But how do I choose which values to use? The integral of sin^2(x) is (x - sinx cosx)/2. Obviously taking cos^(-1)(0) = 3pi/2 or cos^(-1)(0) = pi/2 will give different answers. Now obviously cos^(-1)(x) is typically understood to give an answer in the range [0, pi]. But why wouldn't the integration work with the bound 3pi/2? I am a bit puzzled. Thanks a lot! Isaac P.S. Why not implement LaTeX with this forum?
1. First, if x = cos A [angle should have a different name], then sin A = sqrt(1 - x^2) does not lead directly to -sin^2(A).

2. To be more direct, if x = cos A, then A = cos^(-1)A. So, if x = 0, A = ? If x = sqrt(3)/2, A = ? You need to be familiar already with basic trig functions to do those.

3. Latex does work here.

You should start over by drawing a right triangle with angle A, x on the hypotenuse, and 1 on the adjacent sides. Then sqrt(x^2-1) will be the other side, then take it from there.

 November 27th, 2008, 06:26 PM #3 Newbie   Joined: Nov 2008 Posts: 7 Thanks: 0 Re: Determining bounds when changing variables Thank you for your answer. Obviously I should have written $x=\cos u$. So we have $K=\int_0^{\sqrt 3/2}\sqrt{x^2-1}\: dx = \int_{\cos^{-1}(0)}^{cos^{-1}(\sqrt 3/2)}\sqrt{\sin^2 u}(-\sin u\: du) = -\int_{\cos^{-1}(0)}^{cos^{-1}(\sqrt 3/2)}\sin^2 u \: du = -\left[\frac{x-\sin x \cos x}{2} \right]_{\cos^{-1}(0)}^{cos^{-1}(\sqrt 3/2)}.$ Now obviously the answer will depend on the value chosen for the bounds. My question is, must the bounds be chosen in the range $[0,\pi]$, and, if so, why?
 November 28th, 2008, 08:18 AM #4 Newbie   Joined: Nov 2008 Posts: 16 Thanks: 0 Re: Determining bounds when changing variables if you want to evaluate the integral in terms of u then you need to change the bounds. if you plan on substituting the u's back into x's then you can use the original bounds since the integral was really from x=0 to x=sqrt(3)/2.
November 28th, 2008, 08:48 AM   #5
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Re: Determining bounds when changing variables

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 Originally Posted by bobbyj if you want to evaluate the integral in terms of u then you need to change the bounds. if you plan on substituting the u's back into x's then you can use the original bounds since the integral was really from x=0 to x=sqrt(3)/2.
Thanks for your answer. This is clear to me. What is not exactly clear is why the integral only works when choosing the values of the bounds in the range [0,pi]. Note that this leads to integrating "backwards", from Pi/2 towards Pi/6.

 November 28th, 2008, 11:28 AM #6 Newbie   Joined: Nov 2008 Posts: 16 Thanks: 0 Re: Determining bounds when changing variables I would try to do a substitution involving the identity tan^2(x) = sec^2(x) - 1
 November 28th, 2008, 12:59 PM #7 Senior Member   Joined: Jul 2008 Posts: 895 Thanks: 0 Re: Determining bounds when changing variables Just a note on the bounds, taking you back to those earlier studies as I suggested: Draw a square and cut it in half with a diagonal. You then have a 45,45,90 triangle with sides in the ratio 1:1:sqrt(2). Draw an equilateral triangle, and cut it in half with an altitude. You then have a 30,30,90 triangle with sides in the ratio 1:2:sqrt(3). Label the angles and find the trig ratios of sides to answer the question about what the angles are which are the bounds we are talking about here.

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