Find The Length Of The Curve X=etcos(t), y=etsin(t), 0 ≤ t
Understanding how to determine the length of a curve is a fundamental aspect of calculus, with applications spanning physics, engineering, computer graphics, and more. In this article, we focus on a specific parametric curve defined by the equations \( x = e^t \cos(t) \) and \( y = e^t \sin(t) \), with the parameter \( t \) ranging from a starting point (often 0) to some upper limit. Our goal is to carefully analyze and compute the arc length of this curve, providing a comprehensive guide that delves into the mathematical steps involved, optimization techniques, and practical applications.
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Understanding the Parametric Equations
Before calculating the length of the curve, it’s essential to understand the nature of the parametric equations:
- \( x(t) = e^t \cos(t) \)
- \( y(t) = e^t \sin(t) \)
These equations describe a curve in the xy-plane, parameterized by \( t \), which can be thought of as a "moving" point tracing the path over time or some other variable.
Key Observations About the Curve
- The equations resemble a spiral pattern due to the exponential term \( e^t \) combined with the circular functions \( \cos(t) \) and \( \sin(t) \).
- The point \( (x(t), y(t)) \) can be expressed in polar coordinates, providing a different perspective for analysis.
Expressing the Curve in Polar Coordinates
Given the parametric equations, notice that:
\[
x(t) = e^t \cos(t), \quad y(t) = e^t \sin(t)
\]
which implies:
\[
r(t) = \sqrt{x(t)^2 + y(t)^2} = \sqrt{(e^t \cos t)^2 + (e^t \sin t)^2} = e^t \sqrt{\cos^2 t + \sin^2 t} = e^t
\]
and
\[
\theta(t) = \arctan\left(\frac{y(t)}{x(t)}\right) = \arctan\left(\frac{e^t \sin t}{e^t \cos t}\right) = \arctan(\tan t) = t
\]
since \( \arctan(\tan t) = t \) within principal ranges.
Implication: The curve in polar coordinates is simply \( r = e^t \), with the angle \( \theta = t \). This makes the curve a spiral where the radius grows exponentially with the angle.
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Calculating the Arc Length of the Spiral
The general formula for the length \( L \) of a curve defined in polar coordinates \( r(\theta) \) from \( \theta = \alpha \) to \( \theta = \beta \) is:
\[
L = \int_{\alpha}^{\beta} \sqrt{r(\theta)^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta
\]
In our case, since \( r(t) = e^t \) and \( \theta = t \), we switch variables from \( t \) to \( \theta \). The limits of integration are from \( t = 0 \) to \( t = T \), corresponding to \( \theta = 0 \) to \( \theta = T \).
Step-by-step process:
- Express \( r(\theta) \): \( r(\theta) = e^\theta \)
- Compute \( dr/d\theta \): \( \frac{dr}{d\theta} = e^\theta \)
- Substitute into the arc length formula:
\[
L = \int{0}^{T} \sqrt{e^{2\theta} + (e^{\theta})^2} \, d\theta = \int{0}^{T} \sqrt{e^{2\theta} + e^{2\theta}} \, d\theta
\]
\[
L = \int{0}^{T} \sqrt{2 e^{2\theta}} \, d\theta = \int{0}^{T} \sqrt{2} \, e^{\theta} \, d\theta
\]
- Simplify the integral:
\[
L = \sqrt{2} \int_{0}^{T} e^{\theta} \, d\theta
\]
- Perform the integration:
\[
L = \sqrt{2} \left[ e^{\theta} \right]_0^{T} = \sqrt{2} \left( e^{T} - 1 \right)
\]
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Final Formula for the Length of the Curve
The length \( L \) of the curve from \( t = 0 \) to \( t = T \) is:
\[
\boxed{
L = \sqrt{2} \left( e^{T} - 1 \right)
}
\]
This elegant expression indicates that the length depends exponentially on the upper limit \( T \) of the parameter \( t \).
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Interpretation and Applications
Understanding the length of this spiral has numerous practical applications:
- Physics: Describing the trajectory length of particles moving under specific forces.
- Engineering: Calculating the length of spiral springs or coils.
- Computer Graphics: Rendering curves and paths with precise measurements.
- Mathematics Education: Demonstrating the relationship between parametric and polar representations.
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Optimizing the Calculation for Different Ranges
While we considered \( t \) from 0 to \( T \), the formula can be adapted for other intervals:
- For \( t \) from \( a \) to \( b \):
\[
L = \sqrt{2} \left( e^{b} - e^{a} \right)
\]
- If the interval is infinite (i.e., \( T \to \infty \)), the length diverges, indicating an infinitely long spiral.
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Key Points to Remember
- The parametric equations describe a spiral with exponential growth.
- Converting to polar coordinates simplifies the length calculation.
- The final arc length formula involves an exponential function, illustrating rapid growth.
- The method applies to various ranges and can be adapted to different spirals.
Conclusion
Calculating the length of a curve like \( x = e^t \cos t \), \( y = e^t \sin t \) reveals the beauty of calculus and its ability to handle complex curves with elegant formulas. By understanding the underlying geometry, transforming to polar coordinates, and applying the arc length integral, we derive a concise expression that captures the essence of this exponential spiral’s length. Whether for academic purposes, engineering designs, or computer graphics, mastering these techniques enables precise measurements and deeper insights into the fascinating world of curves.
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Remember: The key to solving such problems lies in recognizing the nature of the parametric equations, leveraging coordinate transformations, and applying integral calculus thoughtfully.
