Calculus help (changing limits of integration)

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Calculus help (changing limits of integration)

Postby SnoringFrog » Wed Sep 14, 2011 5:35 pm

Amazingly, I'm actually kinda understand most of the calc I've done tonight, but this one piece of this problem is stumping me.

I have an integral from 1 to 2. I'm changing from x in the problem to sec(y) (so x = sec(y)). I know that the limits end up being from 0 to pi/3, but I can't figure out how.

I worked it out to this:

2 = sec(y) ; sec^(-1) (2) = y
1 = sec(y) ; sec^(-1) (1) = y
And sec^(-1) would be cos, right? So I have cos(2) = y and cos(1) = y but have no clue how to get to pi/3 and 0 from those.

Can anyone help me out here?


I also tried to just convert back from y to x in the end, but couldn't do that either. My end answer (before evaluating limits) is tan(y) - y, which I confirmed as being correct. But (tan(sec(2)) - 2) - (tan(sec(1)) - 1) didn't get me any closer to sqrt(3) - pi/3 (which is the final answer I'm looking for).
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Postby MomentOfInertia » Wed Sep 14, 2011 7:30 pm

I think you need something along the lines of a

x = tan (y)

y = arctan(x)

type substitution, where y is in radians. Actually it's probably arcsec in this case.

The substitution affects the limits. which is where my memory get kind of fuzzy, it's a direction at least.

I hope that helps.
If you post the original eq. I can probably dig out my textbook and take a crack at it sometime in the next few days.
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Postby SnoringFrog » Wed Sep 14, 2011 8:15 pm

Well, in the next few days I won't care nearly as much. The test is tomorrow XD

Regardless, the eq is this:

integral from 1 to 2 of: (sqrt(x^2 - 1)) / x
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Postby MomentOfInertia » Wed Sep 14, 2011 9:00 pm

x = sec y
y = arcsec x

2 = sec(y) ; sec^(-1) (2) = y
1 = sec(y) ; sec^(-1) (1) = y

substitute:

integral from 1 to 2 of: (sqrt(x^2 - 1)) / x

becomes:

integral from arcsec 1 to arcsec 2 of: [sqrt(sec^2(y) - 1)]/sec y

sec^2(y) -1 is = to tan^2(y) so then we have:

int arcsec 1 to arcsec 2 of: [sqrt{tan^2(y) }]/sec y

Simplifies to:

int arcsec 1 to arcsec 2 of: [tan(y)]/sec y

Or:

int arcsec 1 to arcsec 2 of: [tan(y)[cos(y)]

->

int arcsec 1 to arcsec 2 of: [(sin y)/(cos y)] [cos y]

->

int arcsec 1 to arcsec 2 of: sin (y)

Integrate:

-cos (y) from arcsec 1 to arcsec 2

Or

cos( arcsec 1) - cos( arcsec 2)


Does this help?
If it doesn't I'm gonna have to look up the next bit.

Edit: maybe tomorrow, goodnight. o_o/
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Postby TheMewster » Tue Sep 20, 2011 3:49 pm

[quote="SnoringFrog (post: 1504420)"]Amazingly, I'm actually kinda understand most of the calc I've done tonight, but this one piece of this problem is stumping me.

I have an integral from 1 to 2. I'm changing from x in the problem to sec(y) (so x = sec(y)). I know that the limits end up being from 0 to pi/3, but I can't figure out how.

I worked it out to this:

2 = sec(y) ]

I... wad going to help out here, but that goes way over my head. But if it lightens the mood I once thought sin (from trig) was pronounced sin (like sins in the Bible) and I once said "Well it ought to be a sin for these people to shove this stuff down our throats!!!" Of course I think I've grown too much to make that joke again but still.
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