Problem 71 - Calculus: Early Transcendental Functions

Question

Identify the integrals to which the Fundamental Theorem of Calculus applies; the other integrals are called improper integrals. (a) \(\int_{0}^{4} \frac{1}{x-4} d x\) (b) \(\int_{0}^{1} \sqrt{x} d x\) (c) \(\int_{0}^{1} \ln x d x\)

Step-by-Step Solution

Answer

The integral \(\int_{0}^{4} \frac{1}{x-4} d x\) and \(\int_{0}^{1} \ln x d x\) are improper integrals while \(\int_{0}^{1} \sqrt{x} d x\) can be evaluated using the Fundamental Theorem of Calculus.

Step 1: Identifying the Type of Integral (a)

In the integral \(\int_{0}^{4} \frac{1}{x-4} d x\), the function \(\frac{1}{x-4}\) is not defined at \(x=4\) which is one of the endpoints of the interval of integration, [0, 4]. Therefore, this is an improper integral.

Step 2: Identifying the Type of Integral (b)

In the integral \(\int_{0}^{1} \sqrt{x} d x\), the function \(\sqrt{x}\) is defined and continuous in the interval of integration [0, 1]. Thus, we can apply the Fundamental Theorem of Calculus to evaluate this integral and it is not an improper integral.

Step 3: Identifying the Type of Integral (c)

In the integral \(\int_{0}^{1} \ln x d x\), the function \(\ln x\) is not defined at \(x=0\) which is one of the endpoints of the interval of integration. So, this is an improper integral.

Key Concepts

Improper Integrals

Improper integrals are a fascinating part of calculus. They involve integrals with limits where the function is undefined or unbounded at one or both endpoints of the interval. Such integrals occur in two main cases:

• Case 1: The interval of integration includes a point where the function goes to infinity, like infinity or negative infinity.
• Case 2: The function becomes infinite within the interval or at the boundaries, like having a vertical asymptote in the interval.

For example, in the integral \(\int_{0}^{4} \frac{1}{x-4} dx\), the value \(x=4\) results in division by zero, creating a vertical asymptote, making it improper. Similarly, the integral \(\int_{0}^{1} \ln x dx\) is considered improper because \(\ln x\) is undefined at \(x=0\). To handle these kinds of integrals, calculus employs limits to understand the behavior near the points of undefined behavior. This allows us to still calculate and understand them.

Integral Calculus

Integral calculus is a major branch of calculus focusing on integrals. It studies the accumulation of quantities and the areas under and between curves. Integral calculus is essential for finding quantities like:

• The total accumulated value of a thing over an interval, such as total distance covered.
• The area underneath the curve of a function derived from a graph.

This discipline links directly to differential calculus through the Fundamental Theorem of Calculus, which states that the process of integration (finding of an integral) can be reversed by differentiation. By using antiderivatives, integral calculus can systematically evaluate definite integrals and uncover new insights into functions' behaviors. This branch of calculus is not just for mathematicians—it's used in physics, engineering, statistics, and many other fields to solve real-world problems.

Definite Integrals

Definite integrals play an important role in mathematics, especially in integral calculus. They represent the area under a curve between two specific points on a graph. The notation for a definite integral, \(\int_{a}^{b} f(x) \, dx\), shows:

• \(a\) and \(b\) as the limits of integration, the start and end points.
• \(f(x)\), the integrand, the function under the curve.

The main tool for evaluating definite integrals is the Fundamental Theorem of Calculus, which allows us to find the difference between the values of an antiderivative at two points. A properly defined integrand ensures a defined outcome, which is why not all integrals can be solved using this method.
For example, the integral \(\int_{0}^{1} \sqrt{x} \, dx\) is a typical definite integral that is well-defined and can be solved using this theorem, providing the exact area below \(\sqrt{x}\) from \(x=0\) to \(x=1\). Thus, understanding definite integrals is fundamental to grasp the concept of accumulation and change in numerous scientific and mathematical applications.