What exactly constitutes as an exact value?

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SUMMARY

The discussion centers on the definition of "exact value" in the context of calculus, specifically regarding the calculation of the perimeter of a region bounded by a curve and linear sides. The participant utilized the integral of the square root of the derivative plus one to find the curve's length, resulting in a long irrational number. The instructor required an answer with at least three decimal places, leading to confusion about whether this constituted an approximation or an exact value. The consensus suggests that a closed form expression is considered exact, while a lengthy decimal representation is not.

PREREQUISITES
  • Understanding of calculus concepts, particularly integration and derivatives.
  • Familiarity with perimeter calculations involving curves and linear segments.
  • Knowledge of how to evaluate integrals using a calculator.
  • Concept of exact versus approximate values in mathematical expressions.
NEXT STEPS
  • Research the properties of exact values versus approximations in calculus.
  • Learn about the integral of the square root of the derivative plus one in detail.
  • Study examples of perimeter calculations for regions bounded by curves.
  • Explore closed form expressions and their significance in mathematical solutions.
USEFUL FOR

Students studying calculus, educators teaching mathematical concepts, and anyone interested in the nuances of exact versus approximate values in mathematical evaluations.

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I'm not sure if this is the correct section to post in, but since it relates to calculus and this is more of a general concept than a single problem I figured here would be fine.

On a recent exam we were given a region bounded by a simple curve generated from a function that was bounded from the y-axis to a constant. We were asked to find the exact perimeter of the region. Solving the problem involved solving for the curve and the other two linear sides. Now for the curve I understand you do the integral of the square root of the derivative plus one. I did that, but when I evaluated on the calculator it came out to a very long irrational number. I felt that including a certain number of decimals would be an approximation, not an exact number. So I left it as the value of the other two sides plus the integral. The answer they wanted was anything with at least three decimals. I was not given points as my teacher felt that I did not properly evaluate the problem. Who is the right here?
Sorry if I'm being vague. We are asked not to post the problems/solutions so I was doing my best to be clear without actually giving the problem.
 
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My guess: there is a closed form expression (what you put into the calcuator) and that is what was asked for.
 
If you were give x^2+ y^2= 4 and asked to find the circumference, "4\pi would be an "exact" answer. "12.566370614359172953850573533118" would not be.
 
Okay, thank you to both of you. The answer could not be written out, so I still think I was correct in the notation I used. That being said, I don't make the decisions so I'll just have to live with things.
 

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