Problem 16 - 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. \(\int_{-1}^{1} \frac{1}{5 x^{2}} d x\)

Step-by-Step Solution

Answer

The integral \(\int_{-1}^{1} 1/(5x^2) dx\) is convergent and equals -2/5.

Step 1: Evaluate the Integral

First, integrate the function \(f(x) = 1/(5x^2)\) from \(-1\) to \(1\). The antiderivative of \(f(x)\) is \(-1/(5x)\), so the integral becomes \(-1/(5x)\) evaluated from \(-1\) to \(1\). Hence, \(-1/(5 \cdot 1) - (-1/(5 \cdot -1)) = -1/5 - 1/5 = -2/5.

Step 2: Test for Convergence

While the integral was computed successfully, the value needs to be analyzed for convergence. But since the solution of the integral is a real number, not infinity, this means that the integral is converging.

Step 3: Final Analysis

Based on the results from steps 1 and 2, the integral \(\int_{-1}^{1} 1/(5x^2) dx\) is convergent and equals to -2/5.

Key Concepts

Improper Integrals

Improper integrals are a type of definite integrals where either the interval of integration is infinite, such as \(\int_{a}^{\infty} f(x) dx\), or the function being integrated, \(f(x)\), becomes infinite within the integration interval. In the original exercise, \(\int_{-1}^{1} \frac{1}{5x^{2}} dx\) might seem improper at first glance because the function involves a division by \(x^2\), which could lead to undefined values. However, since \(x^2\) is always positive, this integral does not face the issue of division by zero within the interval from \( -1\) to \(1\), and so it is not an improper integral by definition.

Test for convergence is crucial in improper integrals, as it determines whether the integral yields a finite result or not. The rule of thumb mentioned in the exercise implies that if the limit of \(f(x)\) as \(x\) approaches infinity does not equal zero, the improper integral of \(f(x)\) over \( [a,infinity]\) is divergent. However, if the limit approaches zero, more investigation is needed to confirm convergence.

Antiderivative

The antiderivative, also known as an indefinite integral, of a function \(f(x)\) is another function \(F(x)\) such that \(F'(x) = f(x)\). It is essentially the inverse operation to finding the derivative. When dealing with the integral given in the exercise, \(\int_{-1}^{1} \frac{1}{5x^{2}} dx\), finding the antiderivative is the first step. For the function \(f(x) = \frac{1}{5x^{2}}\), the antiderivative is found by reversing the power rule for derivatives, leading us to \(F(x) = -\frac{1}{5x}\).

This process of finding the antiderivative is essential for evaluating definite integrals from \(a\) to \(b\), where after finding \(F(x)\), we can compute \(F(b) - F(a)\) to find the overall value of the integral.

Limit of a Function

The concept of the limit of a function is crucial in calculus and is particularly important when analyzing the behavior of a function as it approaches a certain point. It is the value that \(f(x)\) approaches as \(x\) approaches a specific value. In the context of improper integrals, if the limit of the function \(f(x)\) approaches a finite number or zero as \(x\) approaches infinity or a point of discontinuity, we can often say that the integral converges to a finite area.

In the exercise provided, while the limit wasn't explicitly discussed, it is implied, since the antiderivative calculation handles the case where \(x\) does not equal zero. Considering limits can help in determining whether an integral is improper and whether it would converge or diverge.

Definite Integrals

Definite integrals represent the net area under the graph of the function \(f(x)\) between two points, \(a\) and \(b\), along the \(x\)-axis. The definite integral is a fundamental concept, as it connects the idea of the antiderivative with the practical application of finding areas. After calculating the antiderivative \(F(x)\), we evaluate it at the upper limit \(b\) and subtract the evaluation at the lower limit \(a\), which is succinctly written as \(F(b) - F(a)\).

In the provided step by step solution, the exercise involves a definite integral, specifically \(\int_{-1}^{1} \frac{1}{5x^{2}} dx\), and the solution follows directly from evaluating the antiderivative from step 1 at the bounds of \( -1\) and \(1\), yielding the result \( -2/5\). This calculation provides us with the area under the curve of \(f(x) = \frac{1}{5x^{2}}\) from \( -1\) to \(1\), which is finite and therefore indicates that the integral converges.