Question

Use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ r=e^{\theta}, \quad 0 \leq \theta \leq \pi $$

Short answer

The approximate length of the curve accurate to two decimal places is 40.17.

Step-by-step solution

Understand the Polar Equation and Plot it

The given polar equation is \(r=e^\theta\), for \(0 \leq \theta \leq \pi\). In a polar graph, each point in the plane is defined by its distance \(r\) from the origin and the angle \(\theta\) from the positive x axis. It would be good to graph this equation using a graphing utility. The graph shows how the distance \(r\) from the origin changes as \(\theta\) increases from 0 to \(\pi\). This polar equation represents a type of spiral shape, where the distance of \(r\) from the origin increases as \(\theta\) increases.

Define the Polar Arc Length Formula

For a curve described by the polar equation \(r(\theta)\), the arc length over the interval \(\alpha \leq \theta \leq \beta\) is given by the following formula:\[ L = \int_{\alpha}^{\beta} \sqrt{[r(\theta)]^2 + [r'(\theta)]^2} d\theta\]where \(r'\) represents the derivative of \(r\) with respect to \(\theta\).

Calculate the Derivative of the Polar Equation

The derivative of the function \(r = e^\theta\) with respect to \(\theta\) is also \(e^\theta\). Thus, the equation becomes:\[ L = \int_{0}^{\pi} \sqrt{(e^\theta)^2 + (e^\theta)^2} d\theta\]

Simplify the Equation

The equation simplifies to:\[ L = \int_{0}^{\pi} e^\theta \sqrt{2} d\theta\]which can be further simplified to:\[ L = \sqrt{2} \int_{0}^{\pi} e^\theta d\theta\]

Integrate the Equation

Finally, integrate the equation. The integral of \(e^\theta\) from 0 to \(\pi\) is \(e^\theta\) evaluated at \(\pi\) and 0.This gives:\[ L = \sqrt{2} *(e^\pi - e^0)\]where \(e^0\) is equal to 1.

Calculate the Numerical Approximation

Calculate the numerical approximation of the value accurate to two decimal places: \[ L = \sqrt{2} *(e^\pi - 1)\]Using a calculator, this approximates to 40.17.