Trigonometric Integral sanity check

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SUMMARY

The discussion centers on the manipulation of the arc length formula, specifically the expression sqrt(1 + tan^2(x)), which simplifies to sqrt(sec^2(x)). The key question raised is whether the square root cancels the sec^2, resulting in sec(x). The conclusion reached is that the square root does not simply yield sec(x) but rather |sec(x)|, emphasizing the importance of considering the absolute value to avoid negative results in arc length calculations. The user confirms their understanding of the bounds of integration, which are from 0 to π/3, ensuring the arc remains in the fourth quadrant.

PREREQUISITES
  • Understanding of trigonometric identities, specifically secant and tangent functions.
  • Knowledge of calculus concepts, particularly arc length and definite integrals.
  • Familiarity with algebraic manipulation involving square roots and absolute values.
  • Basic comprehension of the Cartesian coordinate system and quadrants.
NEXT STEPS
  • Study the properties of absolute values in calculus, particularly in relation to arc length.
  • Explore trigonometric identities and their applications in calculus.
  • Learn about the implications of integrating functions over specific bounds in different quadrants.
  • Review the derivation and applications of the arc length formula in various contexts.
USEFUL FOR

Students studying calculus, particularly those focusing on arc length and trigonometric functions, as well as educators looking to clarify concepts related to integration and algebraic manipulation.

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Homework Statement



Hello,

After some algebraic manipulation my arc length formula of sqrt(1+y')^2) resulted in
sqrt(1+ tan^2 (x))...which can translate into sqrt(sec^2(x)).

My question is does the square root cancel the sec^2 ? becoming sec x?

Like sqrt(x^2) is x?


Thanks!


Homework Equations





The Attempt at a Solution

 
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The sqrt(x^2) isn't x, it's |x|. And there are certainly cases where you can't throw the absolute value away. Be conscious of the sign of the expression you pulled out of the sqrt, or you might get a negative arc length or cancellation between different parts of the arc.
 
Oh I see what you mean, for instance when integrating with respect to the bounds of the definite integral.
I think I am Ok, because my bounds are 0 -> pi/3 and my arc is in the fourth quadrant with respect to x.
Thanks for the answer...and you are absolutely right it is |x|...my mistake
 

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