Problem 47 - Calculus Single Variable

Question

Find an example of a function \(f\), a function \(g\), and values of \(a\) and \(b\) such that $$ \int_{a}^{b} \frac{f(x)}{g(x)} d x \neq \frac{\int_{a}^{b} f(x) d x}{\int_{a}^{b} g(x) d x}. $$

Step-by-Step Solution

Answer

Use \( f(x) = x^2 + 1 \), \( g(x) = x \) from 0 to 1; the result is undefined vs \( \frac{8}{3} \).

Step 1: Understand the Problem

We need to find functions \( f(x) \) and \( g(x) \), along with limits \( a \) and \( b \), such that the integral of the quotient \( \int_{a}^{b} \frac{f(x)}{g(x)} \, dx \) is not equal to the quotient of the integrals \( \frac{\int_{a}^{b} f(x) \, dx}{\int_{a}^{b} g(x) \, dx} \).

Step 2: Select Functions and Values

Let's choose simple functions \( f(x) = x \) and \( g(x) = 1 \). We will use limits \( a = 0 \) and \( b = 1 \).

Step 3: Compute the Integral of the Quotient

We find the integral \( \int_{0}^{1} \frac{x}{1} \, dx = \int_{0}^{1} x \, dx \). This evaluates to \( \left[ \frac{x^2}{2} \right]_{0}^{1} = \frac{1}{2} \).

Step 4: Compute the Quotient of the Integrals

First, evaluate \( \int_{0}^{1} x \, dx = \frac{1}{2} \). Then \( \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 \). Hence, the quotient is \( \frac{1/2}{1} = \frac{1}{2} \).

Step 5: Verify the Inequality

Check if \( \int_{0}^{1} \frac{x}{1} \, dx eq \frac{\int_{0}^{1} x \, dx}{\int_{0}^{1} 1 \, dx} \). Both computations evaluated to \( \frac{1}{2} \), so in this case, they are equal. We need another example where they are not equal.

Step 6: Choose Correct Functions

Let's use \( f(x) = x^2 + 1 \) and \( g(x) = x \) with \( a = 0 \) and \( b = 1 \).

Step 7: Recompute the Integral of the Quotient

Now, calculate \( \int_{0}^{1} \frac{x^2 + 1}{x} \, dx = \int_{0}^{1} (x + \frac{1}{x}) \, dx \), resulting in \( \left[ \frac{x^2}{2} + \ln|x| \right]_{0}^{1} \), which is undefined as the lower limit approaches a nonfinite value.

Step 8: Recompute the Quotient of the Integrals

Now calculate \( \int_{0}^{1} (x^2 + 1) \, dx = \left[ \frac{x^3}{3} + x \right]_{0}^{1} = \frac{4}{3} \) and \( \int_{0}^{1} x \, dx = \frac{1}{2} \). Thus, the quotient is \( \frac{4/3}{1/2} = \frac{8}{3} \).

Step 9: Conclusion

The integral of the quotient is not calculable (undefined) due to the \( \ln|x| \) as \( x \to 0^+ \), while the quotient of the integrals results in \( \frac{8}{3} \). Hence, they are not equal, achieving our goal.

Key Concepts

Quotient of Integrals

In calculus, when dealing with integrals, we encounter various forms and operations, one of them being the quotient of integrals. This scenario occurs when we attempt to integrate a quotient function \( \frac{f(x)}{g(x)} \) across a certain range. But it's important to note that this is different from the quotient of two separate integrals.

• The integral of a quotient is represented as \( \int_{a}^{b} \frac{f(x)}{g(x)} \, dx \).
• Conversely, the quotient of integrals is calculated as \( \frac{\int_{a}^{b} f(x) \, dx}{\int_{a}^{b} g(x) \, dx} \).

While these forms might appear superficially similar, they can lead to very different results, particularly when \( g(x) \) becomes zero at any point in the interval (\(a, b\)), or if it tends towards zero, as this would make the integral of the quotient undefined.
This distinction is often used as a teaching moment in calculus to understand the behavior of integrals.

Function Examples

Choosing appropriate functions is crucial when exploring integrals and their behavior. In our case, understanding the conditions that make \( \int_{a}^{b} \frac{f(x)}{g(x)} \, dx eq \frac{\int_{a}^{b} f(x) \, dx}{\int_{a}^{b} g(x) \, dx} \) requires experimenting with different \( f(x) \) and \( g(x) \).

• A simple example might involve choosing functions like \( f(x) = x \) and \( g(x) = 1 \), which failed to provide a distinction as explained in the solution.
• A more effective pair like \( f(x) = x^2 + 1 \) and \( g(x) = x \) results in an interesting case where the function in the denominator causes an undefined scenario due to its behavior at zero.

This selection demonstrates how different integral properties appear with varying function combinations, and emphasizes the necessity to thoroughly evaluate integration limits to uncover inequalities.

Integral Inequality

Integral inequality is an interesting aspect of calculus. It surfaces when the integral of the quotient differs from the quotient of integrals. This inequality arises due to mathematical properties of functions under integration.

• When \( g(x) \) nears zero within the integral limits, it may cause \( \int_{a}^{b} \frac{f(x)}{g(x)} \, dx \) to become undefined. This is because the denominator cannot be zero as it leads to mathematical uncertainty.
• Simultaneously, the individual integrals of \( f(x) \) and \( g(x) \) may still compute to defined values, leading to distinct results for the quotient when divided.

This inequality forces us to consider the implications of choosing particular regions or functions when analyzing integrals, and it's a grounded reminder of how calculus handles the infinite and the unbounded.

Integration Techniques

There are various integration techniques used to solve problems in calculus. The problem involving the difference between the quotient of integrals and the integral of the quotient requires specific methods.

• Basic integration involves applying fundamental theorems, such as calculating \( \int_{a}^{b} x \, dx \) or \( \int_{a}^{b} 1 \, dx \) using standard formulas for powers and constants.
• For more complex problems, like \( \int_{a}^{b} \frac{x^2 + 1}{x} \, dx \), one might consider substitution or partial fraction decomposition (if applicable), but sometimes, the outcome may show that such integrals are undefined due to logarithmic limits near zero.

Understanding these techniques is crucial for accurate analysis, ensuring the right approach is chosen when evaluating different scenarios, and mastering these can greatly improve one's calculus skills.