Math Problem Statement
Find Length of a Curve
Solution
To calculate the length of the given curve , for ranging from 2 to 3, we use the arc length formula:
Step 1: Differentiate with respect to
The given equation is: Differentiating term by term with respect to :
Step 2: Plug into the arc length formula
The arc length formula becomes:
Step 3: Simplify the expression under the square root
Let: Expanding: So the integrand becomes: Simplify further:
Step 4: Evaluate the integral
This integral does not have a closed-form solution, so it must be computed numerically.
Numerical Solution
Using a numerical integration method (e.g., Simpson's Rule or a calculator):
Let me know if you would like me to show the numerical calculation in more detail!
5 Related Questions
- How is the arc length formula derived?
- How does changing the bounds of affect the curve length?
- Can the length of a curve ever be negative? Why or why not?
- How would the process change if were a function of instead of ?
- What methods exist for solving integrals that don’t have a closed-form solution?
Tip
When solving problems involving arc length, always check whether the integral simplifies to an elementary function or requires numerical approximation.
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Math Problem Analysis
Mathematical Concepts
Calculus
Arc Length of a Curve
Numerical Integration
Formulas
Arc Length Formula: L = ∫ sqrt(1 + (dx/dy)^2) dy
Theorems
Differentiation rules
Integration methods
Suitable Grade Level
Grades 11-12 or early college
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