Question

Find the length of the curve \(x=e^{t}-t, y=4 e^{t / 2},-8 \leq t \leq 3\)

Short answer

The length of the curve must be evaluated numerically due to complexity.

Step-by-step solution

Understand the Curve Formula

The problem involves finding the length of a parametric curve given by the equations: \[ x = e^{t} - t \]\[ y = 4e^{t/2} \]where the parameter \( t \) ranges from \(-8\) to \(3\). The formula for the length of a parametric curve \((x(t), y(t))\) from \(t = a\) to \(t = b\) is:\[ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]

Compute Derivatives

First, compute the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).- For \(x = e^t - t\): \[ \frac{dx}{dt} = e^t - 1 \]- For \(y = 4e^{t/2}\): \[ \frac{dy}{dt} = 4 \cdot \frac{1}{2} \cdot e^{t/2} = 2e^{t/2} \]

Set up the Integral for Curve Length

Substitute the derivatives into the curve length formula:\[ L = \int_{-8}^{3} \sqrt{(e^t - 1)^2 + (2e^{t/2})^2} \, dt \]This integral needs to be simplified and solved.

Simplify the Integrand

Simplify the expression under the square root:\[ \sqrt{(e^t - 1)^2 + (2e^{t/2})^2} = \sqrt{(e^t - 1)^2 + 4e^t} \]Notice that this may require substitution or numerical approaches to solve efficiently.

Evaluate the Integral

Since the simplified form does not lend itself easily to analytical integration, use numerical methods or a computational tool to evaluate:\[ L = \int_{-8}^{3} \sqrt{(e^t - 1)^2 + 4e^t} \, dt \]Common numerical integration techniques include Simpson's Rule or the Trapezoidal Rule, or using a calculator or software that supports numerical integration.

Key concepts

Understanding Curve Length

When we talk about the length of a curve, we're essentially discussing how far we would travel if we walked along the curve's path, from one point to another. For curves described using parametric equations, the length is found using a specific formula. This mathematical formula takes into account the derivatives of the curve's functions with respect to a parameter, usually denoted as \( t \).
The formula for finding the length \( L \) of a parametric curve \((x(t), y(t))\) over the parameter interval \([a, b]\) is: \[ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]
This equation combines the rate of change in the horizontal direction, \( \frac{dx}{dt} \), with the rate of change in the vertical direction, \( \frac{dy}{dt} \).

• The square root expression \( \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \) effectively calculates the infinitesimal "straight-line" distances along the curve.
• The integral then sums these distances from the start to the end of the interval \([a, b]\).

The Role of Derivatives

Derivatives play a crucial role in determining the curve's length because they describe how the curve moves—meaning they reflect the curve's slope and shape. For each small step along the curve, the derivatives indicate how much the curve shifts horizontally and vertically.
To find the length of our given curve \( x = e^t - t, y = 4e^{t/2} \), we first compute the derivatives with respect to \( t \).

• For \( x(t) = e^t - t \), the derivative \( \frac{dx}{dt} \) is \( e^t - 1 \), which represents the rate of change in the x-direction.
• For \( y(t) = 4e^{t/2} \), the derivative \( \frac{dy}{dt} \) is \( 2e^{t/2} \), showing how the y value changes as \( t \) changes.

These derivatives are then plugged into the curve length formula, which helps calculate the total length of the curve by considering these changes simultaneously.

Utilizing Numerical Integration

Calculating the integral for the length of a complex curve often requires numerical integration. This is especially true when the integrand is tricky or does not have a straightforward antiderivative.
The integral we need to solve in this exercise is: \[ \int_{-8}^{3} \sqrt{(e^t - 1)^2 + 4e^t} \, dt \]

• This expression doesn't simplify easily into a form that can be solved by basic calculus, making numerical methods necessary.
• Numerical integration techniques, such as Simpson's Rule or the Trapezoidal Rule, approximate the integral by dividing the interval into smaller segments and calculating the area under the curve for these segments.
• Modern computational tools, like calculators and software programs, provide effective means to handle such calculations, yielding accurate results through these numerical approaches.

Using these methods allows us to find an approximate value for the length of the curve, which is often sufficient for practical applications.