How to Find the Exact Length of a Curve by Antidifferentiation?

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SUMMARY

The discussion focuses on finding the exact length of a curve defined by the integral of the function \(\sqrt{9t^2 + 6t}\) from 0 to \(y\) for \(1 < y < 5\). The length of the curve is calculated using the formula for arc length, which involves integrating \(\sqrt{1 + (dx/dy)^2}\) from 1 to 5. The derivative \(dx/dy\) is determined to be \(9y^2 + 6y\), leading to the expression \(18y + 6\) for the arc length calculation. The user encountered a domain error while attempting to compute this in a calculator, indicating a potential issue with the input values or the function's domain.

PREREQUISITES
  • Understanding of integral calculus and the Fundamental Theorem of Calculus
  • Familiarity with the arc length formula in calculus
  • Knowledge of derivatives and their application in curve analysis
  • Basic proficiency in using graphing calculators for calculus problems
NEXT STEPS
  • Learn how to apply the arc length formula for different types of curves
  • Study the Fundamental Theorem of Calculus in depth
  • Explore common domain errors in calculus and how to troubleshoot them
  • Practice antidifferentiation techniques with various functions
USEFUL FOR

Students studying calculus, particularly those focusing on integral calculus and arc length problems, as well as educators looking for examples of applying the Fundamental Theorem of Calculus in real-world scenarios.

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URGENT! Find exact length of curve!

Homework Statement


Sorry I don't know how to type the integral symbol... But here is the question!

A curve is given by x= the integral from 0 to y of [(9t2+6t)^1/2] dt for 1< y < 5. Find the exact length of the curve analytically by antidifferentiation.

Homework Equations


The length of a curve equation is th integral from a to b of the square root of [1+ (dy/dx)2]


The Attempt at a Solution


I know that the derivative of the integral of the curve is equal to [9y2+6y], becuase of the fundamental theorum of calculus. Then I find the derivative of that which is 18y+6, and I plugged it into the curve length equation. But when I tried putting it into my calculator it had a domain error... why!?

All help is appreciated, I need to know this for my mid term tomorrow. Thank you!
 
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Your are going way too fast here. Take it step by step. Since you are given x as a function of y, you want to integrate sqrt(1+(dx/dy)^2)*dy from 1 to 5. What's dx/dy again?
 

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