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

In summary, the problem is to find the exact length of a curve represented by x = the integral from 0 to y of [(9t^2+6t)^1/2] dt for 1 < y < 5. The solution involves using the fundamental theorem of calculus to find the derivative of the integral, which is equal to [9y^2+6y]. However, when attempting to plug this into the curve length equation, there was a domain error. The next step is to take the derivative of [9y^2+6y] and plug it into the curve length equation, integrating from 1 to 5 to find the exact length of the curve.
  • #1
mark8623
1
0
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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  • #2


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?
 

1. What is the purpose of finding the exact length of a curve?

The purpose of finding the exact length of a curve is to accurately measure the distance between two points on a curved line. This is important in various fields such as mathematics, engineering, and physics where precise measurements are necessary for calculations and designs.

2. What factors affect the length of a curve?

The length of a curve is affected by the curvature of the line, the scale at which it is measured, and the starting and ending points chosen for measurement. Additionally, the type of curve (e.g. straight, circular, parabolic) and the equation used to define the curve also play a role in determining its length.

3. How do you calculate the exact length of a curve?

The exact length of a curve can be calculated using a mathematical technique called integration. This involves breaking the curve into infinitesimally small straight segments and summing their lengths using calculus. Alternatively, for simpler curves, there are specific formulas and methods that can be used to find the length without using integration.

4. Can the exact length of a curve ever be determined?

Yes, the exact length of a curve can be determined using mathematical methods. However, due to the infinite number of points on a curve, the exact length may be a theoretical concept rather than a practical one. In real-world applications, a close approximation of the exact length is often sufficient.

5. Are there any tools or software that can help find the exact length of a curve?

Yes, there are various tools and software available that can help find the exact length of a curve. These include graphing calculators, computer programs, and online calculators that use integration or other methods to calculate the length. Additionally, there are specialized software programs used in fields such as engineering and architecture that have built-in functions for finding the length of curves.

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