Problem 49 - Calculus Single Variable

Question

A definite integral is given. Do not attempt to calculate its value \(V .\) Instead, find the extreme values of the integrand on the interval of integration, and use these extreme values together with the inequalities of line (5.3.5) to obtain numbers \(A\) and \(B\) such that \(A \leq V \leq B\). $$ \int_{1}^{4} \frac{1}{x^{2}-4 x+5} d x $$

Step-by-Step Solution

Answer

The range of the integral is \( \frac{3}{5} \leq V \leq 3 \).

Step 1: Identify the Integrand

The integrand is the function inside the integral: \( \frac{1}{x^{2}-4x+5} \). Our goal is to find the extreme values for this function over the interval \([1, 4]\).

Step 2: Find the Derivative of the Integrand

First, find the derivative of the integrand, \( f(x) = \frac{1}{x^{2} - 4x + 5} \). Use the quotient rule: \[ f'(x) = \frac{0 \cdot (x^2 - 4x + 5) - (1)(2x - 4)}{(x^2 - 4x + 5)^2} = \frac{-2x + 4}{(x^2 - 4x + 5)^2} \].

Step 3: Critical Points and Extreme Values

Set the derivative \( f'(x) = 0 \) to find critical points: \(-2x + 4 = 0\). Solving gives \( x = 2 \). Evaluate \( f(x) \) at the endpoints \( x = 1, 4 \) and at \( x = 2 \). The function values are \( f(1) = 1 \), \( f(2) = \frac{1}{1} = 1 \), \( f(4) = \frac{1}{5} \).

Step 4: Determine the Extreme Values

The extreme values on \([1, 4]\) are \( f(2) = 1 \) as the maximum and \( f(4) = \frac{1}{5} \) as the minimum. Ensure there are no higher or lower values in the interval by analyzing the behavior of the function at these points and considering if the function is increasing or decreasing.

Step 5: Using the Definite Integral Inequality

Use the inequality for integrals: if \( m \leq f(x) \leq M \) for \( x \in [a, b] \), then \( m(b-a) \leq \int_{a}^{b} f(x) \, dx \leq M(b-a) \). Here, \( m = \frac{1}{5}, \ M = 1, \ b-a = 3 \). Thus, \( \frac{3}{5} \leq \int_{1}^{4} \frac{1}{x^{2}-4x+5} \, dx \leq 3 \).

Key Concepts

Definite Integral

A definite integral represents the area under a curve between two specific points on the x-axis. It has fixed upper and lower bounds, known as the limits of integration. For the integral \[\int_{1}^{4} \frac{1}{x^2 - 4x + 5} \, dx\]the bounds are 1 and 4. The goal with definite integrals is to find the total area under the curve of the function from the lower bound to the upper bound. This concept ensures that we have a set interval over which we're evaluating the integral, leading to a specific numeric value rather than a function.

In many scenarios, as shown in the exercise, you don't need to compute the exact area directly. Instead, you can relate the integral's value to certain inequalities by determining the extreme values of the integrand (the function inside the integral) over the given interval. This approach allows us to estimate a range for the integral's value, providing insight without needing an exact calculation.

Extreme Values

Extreme values refer to the highest (maximum) and lowest (minimum) points on a curve within a specific interval. These values occur where the derivative of a function equals zero or is undefined—critical points. In the exercise, we analyze the function\[f(x) = \frac{1}{x^2 - 4x + 5}\]over the interval \([1, 4]\).

By finding where the derivative equals zero, we obtained the critical point at \(x = 2\). Evaluating the integrand at the critical point and the endpoints, we found:

• \(f(1) = 1\)
• \(f(2) = 1\)
• \(f(4) = \frac{1}{5}\)

Thus, the extreme values are 1 and \(\frac{1}{5}\). Knowing these values allows us to apply inequalities to bound the definite integral's value, providing reliable estimates for the area calculated by the integral.

Quotient Rule

The quotient rule is a technique for differentiating functions that are ratios of two other functions. If you have a function in the form \( \frac{u(x)}{v(x)} \), the derivative \(f'(x)\) is calculated by\[f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}\]This rule helps when dealing with complex fractions that contain differentiable functions in both the numerator and the denominator.

In our exercise, we used the quotient rule to find the derivative of the integrand \(f(x) = \frac{1}{x^2 - 4x + 5}\). Applying the quotient rule:

• The numerator derivative: 0 (as the numerator is a constant 1).
• The denominator derivative: \(2x - 4\).
• The resultant derivative: \(f'(x) = \frac{-2x + 4}{(x^2 - 4x + 5)^2}\)

Using this derivative, we pinpointed critical points for identifying extreme values, which further helped in bounding the definite integral. The quotient rule provides a systematic way to handle these derivatives, crucial for analyzing the behavior of functions within the given bounds.