Math Problem Statement
A curve is given by the following parametric equations. X(T)=9tsin(t), y(t)=9tcos(t). Find the arc length, S, of the curve over the interval [0,1]. Enter the exact value with no decimals
Solution
To find the arc length of a curve given by parametric equations and , we use the following formula for parametric curves:
For the given parametric equations:
Step 1: Differentiate and
Step 2: Compute the square of the derivatives
Step 3: Simplify the sum
Now, we add these two expressions:
Expand the terms inside:
Adding these two results gives:
Thus, we have:
Step 4: Integrate to find the arc length
Now, we can find the arc length by integrating:
This integral is a standard form, and its solution is:
Step 5: Evaluate the definite integral
Now, evaluate the integral from to :
Simplifying:
Thus, the exact arc length is:
Final Answer:
The arc length is:
Would you like more details or have any questions?
Related Questions:
- How does parametric differentiation apply in other contexts like velocity or acceleration?
- What are the common applications of arc length in physics or engineering?
- How would the integral change if the parameterization involved polar coordinates?
- Can you find the arc length of a simpler parametric curve like , ?
- What are the challenges in solving integrals involving square roots, and how can they be overcome?
Tip: Understanding parametric curves helps in analyzing complex motion, especially in multi-dimensional spaces.
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Math Problem Analysis
Mathematical Concepts
Calculus
Parametric Equations
Arc Length
Formulas
Arc length formula for parametric curves: S = ∫[a, b] sqrt((dx/dt)^2 + (dy/dt)^2) dt
Derivative of parametric functions: dx/dt, dy/dt
Theorems
Pythagorean Identity: sin^2(t) + cos^2(t) = 1
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12, College Level
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