Problem 38 - Applied Calculus

Question

In this problem, you will show that the following improper integral converges to 1 . $$\int_{1}^{\infty} \frac{1}{x^{2}} d x$$ (a) Use the Fundamental Theorem to find \(\int_{1}^{b} 1 / x^{2} d x\). Your answer will contain \(b\). (b) Now take the limit as \(b \rightarrow \infty\). What does this tell you about the improper integral?

Step-by-Step Solution

Answer

The improper integral converges to 1.

Step 1: Determine the Indefinite Integral

First, we need to find the antiderivative of the function \( f(x) = \frac{1}{x^2} \). Integrating \( \frac{1}{x^2} \) gives us \( f(x) = \int x^{-2} \, dx = -x^{-1} + C = -\frac{1}{x} + C \), where \( C \) is the constant of integration.

Step 2: Evaluate the Definite Integral from 1 to b

Next, apply the Fundamental Theorem of Calculus to evaluate \( \int_{1}^{b} \frac{1}{x^2} \, dx \). Using the antiderivative from Step 1, we have: \[\left[-\frac{1}{x}\right]_1^b = \left(-\frac{1}{b}\right) - \left(-\frac{1}{1}\right) = -\frac{1}{b} + 1.\] This expression is the definite integral from 1 to \( b \).

Step 3: Take the Limit as b Approaches Infinity

Now, take the limit of the expression obtained in Step 2 as \( b \to \infty \):\[\lim_{{b \to \infty}} \left(-\frac{1}{b} + 1\right) = \lim_{{b \to \infty}} (-\frac{1}{b}) + 1 = 0 + 1 = 1.\] This shows that the improper integral \( \int_{1}^{\infty} \frac{1}{x^2} \, dx \) converges to 1.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a powerful tool that links the concept of differentiation with integration. This theorem enables us to evaluate definite integrals using antiderivatives. It states that if a function is continuous on a closed interval \([a, b]\) and \({F}\) is an antiderivative of \({f}\) on \([a, b]\), then:

• \[ \int_a^b f(x) \, dx = F(b) - F(a) \]

This theorem simplifies the process of calculating areas under curves. Instead of summing infinitely small slices, we use antiderivatives to quickly find the answer. Knowing this theorem well is crucial. It transforms the integral problem into simply evaluating the antiderivative function at the boundaries, \({b}\) and \({a}\). In our specific problem, once we found the antiderivative of \(\frac{1}{x^2}\), we applied this theorem between 1 and \({b}\) to find the definite integral, which was \(-\frac{1}{b} + 1\).

antiderivative

An antiderivative is a function which reverses differentiation. In simple terms, it is the opposite operation of taking a derivative. For example, if you calculate a derivative of a function and then find the antiderivative, you should end up back at the original function, plus a constant. For the function \(f(x) = \frac{1}{x^2}\), finding the antiderivative is crucial.Finding an antiderivative is a fundamental step when solving integrals. The antiderivative of \(x^{-2}\) is obtained by reversing the power rule for derivatives. Thus, \(\int x^{-2} \, dx = -x^{-1} + C\), which simplifies to \(-\frac{1}{x} + C\).This antiderivative helps in determining the integral over an interval, forming the basis to apply the Fundamental Theorem of Calculus. Remember, every function has infinitely many antiderivatives, differing by a constant, denoted by \(C\).

limit of a function

The concept of the limit of a function is crucial when dealing with improper integrals. A limit describes the value that a function approaches as the input approaches some value. In mathematics, when expressing a limit as \({x}\) approaches a particular number \({a}\) or even infinity, it helps predict the behavior of the function at that input.For an improper integral like \[\int_{1}^{\infty} \frac{1}{x^2} \, dx\], calculating the integral involves taking the limit of the evaluated antiderivative as the upper bound \({b}\) approaches infinity. Here, the definite integral becomes \(-\frac{1}{b} + 1\), and

• we calculate \(\lim_{b \to \infty} (-\frac{1}{b} + 1) = 1 \)

Using limits in this way shows that even though the region of integration extends infinitely, the integral converges to a finite value, 1 in this case. Understanding limits as \(x\) moves towards infinity is essential to solve improper integrals effectively.