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. $$ r=\sec \theta, \quad 0 \leq \theta \leq \frac{\pi}{3} $$

Short answer

The length of the curve for the given polar equation and interval can be approximated using a graphing utility to first graph the function and then compute the integral representing the arc length.

Step-by-step solution

Graph the polar equation

In the graphing utility, input the polar equation \(r=\sec \theta\). Make sure the settings are in polar and not in cartesian. Insert the limits for the angle \(0 \leq \theta \leq \frac{\pi}{3}\) and graph the equation.

Use arc length formula for polar coordinates

The formula for approximate length of a curve in polar coordinates is given by \[L= \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2 } d\theta \]. In our case, the \(r = \sec \theta\) and \( \frac{dr}{d\theta} = \sec \theta \tan \theta \). Plug these in the formula to get \[L= \int_0^{\frac{\pi}{3}} \sqrt{\sec^2 \theta + \left(\sec \theta \tan \theta \right)^2 } d\theta\]

Use graphing utility to approximate the length of the curve

In the graphing utility, select the 'integral' or 'integrate' function. Input the above integral and compute the integral from the lower limit \(0\) to the upper limit \(\frac{\pi}{3}\). The graphing utility will then calculate the approximate length of the curve.

Key concepts

Graphing Polar Equations

When it comes to understanding curves and shapes in coordinate geometry, one often encounters the cartesian coordinate system. However, another valuable system for graphing equations is the polar coordinate system, which is especially useful for circular and spiral shapes.

Graphing polar equations involves plotting points and curves on a grid where each point is determined by a distance from the origin (the radius r) and an angle θ from the positive x-axis. Unlike cartesian coordinates that use (x, y) pairs, polar coordinates are denoted as (r, θ).

To graph the polar equation r = sec θ, one must first understand that the secant function is the reciprocal of cosine. Thus, it may approach infinity when cosine approaches zero - this typically results in a curve known as a secant spiral. To graph it, we set our calculator or graphing utility to polar mode and input our function, also specifying the interval for θ from 0 to π/3. The resulting graph shows how the curve behaves within this specific range, allowing a visual understanding of the curve's properties before proceeding to length calculations or integrations.

Using graphing utilities not only helps visualize these equations but also enables a more straightforward approach to solving trigonometric values that might be complex to compute by hand.

Polar Equations Integration

Integration in the context of polar equations is a method used to calculate various properties, including area and arc length. The integration of polar functions can be more complex than cartesian functions because the integrand typically includes trigonometric functions, as in the exercise we are looking at.

The integral we use to find the length of a curve in polar coordinates includes both the function r and its derivative dr/dθ. For the given equation r = sec θ, the derivative dr/dθ is sec θ tan θ. This derivative is then plugged into our arc length formula: \[ L= \int_a^b \sqrt{r^2 + \left(\frac{dr}{dθ}\right)^2 } dθ \]

After determining the correct form of the integral, using a graphing utility's integration capabilities simplifies the process of finding an approximation of the curve's length. This is particularly helpful for complicated integrals that are difficult to solve analytically and require numerical approximation techniques.

Curve Length Approximation

The arc length of a curve in polar coordinates can be a challenging concept because it isn't as straightforward as measuring a straight line distance. The length of a curve is essentially the distance one would travel along the curve, which in polar coordinates must take into account changes in both radius and angle.

The approximation for the length of a curve defined by a polar equation is given by the integral formula already discussed. To apply this in practice, as seen in the step by step solution, we evaluate this integral over the interval from 0 to π/3 for our specific function r = sec θ. It is integral to understand that this formula encapsulates the essence of differential calculus - it sums up an infinite number of infinitesimally small linear segments along the curve to approximate its total length.

One thing to note for students is that the accuracy of approximation relies heavily on the capabilities of the graphing utility used. Higher-tier graphing tools may provide a more accurate result due to better numerical integration algorithms. Understanding this concept enables students to tackle similar problems with other polar functions and reinforces the connection between the geometric representation of equations and their corresponding analytical methods.