Question

True or False? In Exercises 55-60, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{a}^{b} f(x) g(x) d x=\left[\int_{a}^{b} f(x) d x\right]\left[\int_{a}^{b} g(x) d x\right] $$

Short answer

The statement is false. The integral of the product of two functions is not equal to the product of their individual integrals, as demonstrated by the counterexample.

Step-by-step solution

Analyze the statement

Consider the statement given, which is \[\int_{{a}}^{{b}} f(x) g(x) dx = \left[\int_{{a}}^{{b}} f(x) dx\right] \left[\int_{{a}}^{{b}} g(x) dx\right]\] This implies the integral of the product of two functions is the product of their individual integrals.

Disprove by counterexample

Let's disprove this by taking a counterexample. Let's take \(f(x) = x\) and \(g(x) = x\). According to the above statement \[\int_{{a}}^{{b}} x^2 dx = \left[\int_{{a}}^{{b}} x dx\right] \left[\int_{{a}}^{{b}} x dx\right]\]Calculate each side to confirm the equality.

Calculate Left-Hand Side

Calculate the left-hand side of the equation, which is the integration of \(x^2\):\[\int_{{a}}^{{b}} x^2 dx = \frac{1}{3}b^3 - \frac{1}{3}a^3\]

Calculate Right-Hand Side

Calculate the right-hand side of the equation, which is the square of the integration of \(x\):\[\left[\int_{{a}}^{{b}} x dx\right] \left[\int_{{a}}^{{b}} x dx\right] = \left[\frac{1}{2}b^2 - \frac{1}{2}a^2\right]^2\]

Compare the Results

As we compare the left-hand and right-hand side, we see that they are not equal in general. Hence, the statement is false. The integral of the product of two functions is not equal to the product of their individual integrals.

Key concepts

Calculus

Calculus is a branch of mathematics that focuses on studying change and motion. Using the tools of calculus, we can analyze the behavior of functions and understand how variables evolve over time.

Integral calculus, the area we're exploring here, is concerned with accumulation and area. It is used to find quantities like areas under curves, total accumulated change, and volumes of objects. This concept requires a good grasp of fundamental properties involving limits and infinite sums that allow us to solve problems in physics, engineering, economics, and beyond. The main two operations in calculus are differentiation and integration, which are inverse processes.

Integration

Integration is one of the two key operations in calculus, alongside differentiation. It can be seen as the process of calculating the area under a graph of a function, also known as the integral of the function over an interval. Integrals are classified into two types: indefinite integrals, which represent a family of functions (also known as antiderivatives), and definite integrals, which evaluate to a specific number representing the total area under the curve between two points on the function's graph.

When you integrate a function, you're essentially adding up an infinite number of infinitesimal quantities to calculate a total. It's an indispensable tool in finding quantities like displacement from velocity, and it's fundamental to understanding the accumulation of quantities over time.

Counterexample

In mathematics, a counterexample is a specific case or example that disproves a hypothesis or statement. For any given statement, if a counterexample exists, then the statement cannot be universally true. This concept is particularly useful when a statement seems plausible, or it's challenging to prove directly that it's false.

In the context of integrals, providing a counterexample can quickly demonstrate that an assumed property is not valid, such as the false notion that the integral of a product is equal to the product of the integrals. Finding and understanding counterexamples helps students to deepen their understanding of mathematical concepts by exploring their limitations and clarifying misunderstandings.

Integral Properties

Integral properties are rules that help in simplifying the integration process or solving integrals more effectively. Some of the most fundamental properties include the linearity of the integral, which allows us to integrate terms separately and pull out constant factors; the additive property, permitting the integral over an interval to be split into the sum of integrals over subintervals; and the property related to odd and even functions, aiding in simplifying definite integrals over symmetric intervals.

However, it's crucial to understand these properties do not always apply in more complex scenarios, such as when dealing with the product of functions. Misapplying these properties, as the exercise shows, can lead to incorrect results. Deep exploration of these properties can help students avoid mistakes and better understand the nuances and limits of integration techniques.