Problem 33 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\pi} \tan x d x\)

Step-by-Step Solution

Answer

The integral \(\int_{0}^{\pi} \tan x d x\) is divergent.

Step 1: Evaluate the limit of the function

First, we evaluate the limit of the function \(\tan x\) as \(x \rightarrow \pi\) from the left, denoted as \(\lim_{x \rightarrow \pi^-} \tan x\). This limit translations into evaluating the value of tangent function slightly before \(\pi\), where it is undefined.

Step 2: Determine if the integral diverges or converges

As tangent approaches \(\pi\) from the left, the function goes to negative infinity. Since the limit is not zero, \(\lim_{x \rightarrow \pi^-} \tan x = -\infty\), we can say using the given criteria that the integral \(\int_{0}^{\pi} \tan x d x\) diverges.

Key Concepts

Improper Integrals

Improper integrals are a special type of integral that deal with unbounded intervals or unbounded functions. These integrals need extra attention because they involve limits to deal with their infinite aspects. For instance, an integral might have limits going towards infinity or a function that becomes infinite at certain points within the interval. Instead of using usual integration techniques, we must carefully compute them by taking limits.

Consider the integral \[ \int_{a}^{b} f(x)\, dx \] If the limit is either infinite or if there is a point within \[a, b\] where \(f(x)\) approaches infinity, the integral becomes improper.

Improper integrals can still be handled meaningfully by determining if they converge or diverge. They converge if the area under the curve is finite, otherwise, they diverge. This involves evaluating the limit as the function or bounds approach infinity.

Convergence and Divergence

When dealing with improper integrals, two key terms, convergence and divergence, come into play. They essentially describe the behavior of the integral as it approaches potentially problematic points.

Convergence means that as we calculate the integral and resolve the infinity issues, the result is a finite number. This indicates that overall, the area under the curve represented by the function is limited, even when stretching towards infinity.

• A convergent improper integral comes to a specific number.
• Convergence implies manageable behavior over infinite stretches.

Divergence, on the other hand, indicates that the integral does not settle to a finite limit. In such cases, the area under the curve is infinite or undefined. This is often due to a section of the function going to infinity or the interval being infinitely large.

• A divergent improper integral does not yield a finite result.
• Divergence means unmanageable or limitless behavior over infinite stretches.

In the example of \(\int_{0}^{\pi} \tan x\, dx \), as \(x\) approaches \(\pi\) from the left, \(\tan x\) tends to \(-\infty\), making the integral divergent.

Tangent Function

The tangent function, \( \tan x \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine function to the cosine function:

\[ \tan x = \frac{\sin x}{\cos x} \]The properties of the tangent function can lead to interesting behaviors that play an important role in calculus.

• The function has a periodic nature with a period of \(\pi\).
• Tangent is undefined when \(\cos x = 0\). This happens at odd multiples of \(\frac{\pi}{2}\).
• The function approaches infinity as \(x\) approaches these undefined points from one side, and negative infinity from the other side, creating vertical asymptotes.

In calculus, understanding where \( \tan x \) is undefined is key when evaluating integrals like \(\int_0^\pi \tan x \, dx\). Since \(\tan x\) becomes undefined at \(x = \pi\), this directly influences the convergence or divergence of integrals related to this function.