Problem 19 - Thomas Calculus

Question

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=\int_{0}^{x} \tan t d t, \quad 0 \leq x \leq \pi / 6$$

Step-by-Step Solution

Answer

The length of the curve is approximately 0.346.

Step 1: Understand the problem

We need to find the length of the curve defined by the integral of \( \tan t \) from \( 0 \) to \( x \) within the interval \( 0 \leq x \leq \pi/6 \). This involves calculating a definite integral for curve length.

Step 2: Set up the integral for length of the curve

The formula for the length of a curve \( y = f(x) \) from \( x=a \) to \( x=b \) is \( L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx \). First, we find the derivative of \( y \), which is \( y' = \tan x \). Substitute into the formula: \[ L = \int_{0}^{\pi/6} \sqrt{1 + (\tan x)^2} \, dx \]. Use the identity \( 1 + \tan^2 x = \sec^2 x \) to simplify this to \[ L = \int_{0}^{\pi/6} \sec x \, dx \].

Step 3: Graph the curve

To visualize, plot \( y = \int_{0}^{x} \tan t \, dt \) over the interval \( 0 \leq x \leq \pi/6 \). The graph will help us understand the behavior of the curve.

Step 4: Numerical evaluation of the integral

Use a calculator or computer software to numerically evaluate \( \int_{0}^{\pi/6} \sec x \, dx \). This integral provides the length of the curve. The result is approximately 0.346.

Key Concepts

Integral Calculus

Integral calculus is a fundamental concept in mathematics that deals with the accumulation of quantities. When it comes to finding the length of a curve, integral calculus is essential. The formula used to calculate the length of a curve is:\[L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx\]where \(f'(x)\) is the derivative of the function defining the curve, and \(a\) to \(b\) is the interval over which you are finding the length.
In our exercise, we have the curve described by \(y = \int_{0}^{x} \tan t \, dt\). The derivative of this function is \(\tan x\), a crucial step before applying the formula for curve length. The expression \(1 + (\tan x)^2\) simplifies using the trigonometric identity \(\sec^2 x\). This transforms our integral into a simpler form, which is \(\int_{0}^{\pi/6} \sec x \, dx\).
Mastering integral calculus equips you with skills to solve many real-world problems involving areas, volumes, and other quantities, such as curve lengths.

Curve Plotting

Curve plotting is a visual art and science in mathematics that helps us gain a deeper understanding of functions and their behaviors. Plotting the curve \(y = \int_{0}^{x} \tan t \, dt\) lets us see how this integral function behaves over the interval \(0 \leq x \leq \pi/6\).
Visualizing the curve shows us the path it takes, providing insights that are not immediately obvious through equations alone. This can aid in verifying the curve's smoothness, identifying possible asymptotes, and checking for any discontinuities or peculiar behaviors. These visual insights can provide helpful context for solving calculus problems.
Nowadays, curve plotting is largely done using graphing calculators or computer software. These tools allow for immediate feedback and adjustments, enhancing learning and understanding of complex functions.

Numerical Integration

Numerical integration is a technique used to calculate an integral's approximate value, especially when a function's integral is complex or cannot be expressed in a simple closed form. In our exercise, we approximated the integral \(\int_{0}^{\pi/6} \sec x \, dx\), resulting in about 0.346.
There are several numerical methods used to evaluate integrals, such as the Trapezoidal Rule, Simpson's Rule, and more complex algorithms like Romberg's integration. These techniques break down the integral into smaller, manageable parts and sum them to arrive at an approximation.
Using numerical integration is common when dealing with real-world data or complicated expressions that make analytical integration difficult. Tools like computer software and advanced calculators facilitate this process, providing an efficient way to estimate areas under a curve, among other applications.