Question

If the change of variables \(u=x^{2}-4\) is used to evaluate the definite integral \(\int_{2}^{4} f(x) d x,\) what are the new limits of integration?

Short answer

Question: Using the change of variables \(u = x^2 - 4\), find the new limits of integration for the definite integral \(\int_{2}^{4} f(x) dx\). Answer: The new limits of integration are \(u = 0\) (lower limit) and \(u = 12\) (upper limit).

Step-by-step solution

Identify the change of variables

The given change of variables is \(u = x^2 - 4\). We need to evaluate the new limits of integration for the definite integral \(\int_{2}^{4} f(x) dx\) using this change of variables.

Find the new lower limit of integration

To find the new lower limit of integration, substitute the original lower limit \(x = 2\) into the change of variables equation \(u = x^2 - 4\): \(u = (2)^2 - 4\) \(u = 4 - 4\) \(u = 0\) So, the new lower limit of integration is \(u = 0\).

Find the new upper limit of integration

To find the new upper limit of integration, substitute the original upper limit \(x = 4\) into the change of variables equation \(u = x^2 - 4\): \(u = (4)^2 - 4\) \(u = 16 - 4\) \(u = 12\) So, the new upper limit of integration is \(u = 12\).

Write the new limits of integration

The new limits of integration after using the change of variables \(u = x^2 - 4\) to evaluate the definite integral \(\int_{2}^{4} f(x) dx\) are: Lower limit: \(u = 0\) Upper limit: \(u = 12\)

Key concepts

Change of Variables

When we deal with definite integrals, sometimes integrating directly with the original variable can be challenging. This is where the concept of a 'change of variables' comes into play, also known as a substitution method. It's a technique in calculus that simplifies the integration process by transforming a difficult integral into a more manageable form.

A change of variables involves introducing a new variable, typically denoted as 'u', that replaces the original variable, in this case, 'x'. The relationship between 'u' and 'x' is established via a substitution equation. For instance, consider the substitution equation used in the exercise: \( u = x^2 - 4 \).

This equation describes how each 'x' value relates to a 'u' value. Our task is to rewrite the integral in terms of this new variable. The essential benefits include simplifying the integrand and potentially making the integral more straightforward to evaluate. For successful change of variables, one must also modify the limits of integration which we'll discuss in the next section.

Limits of Integration

When applying the change of variables to a definite integral, the original limits of integration in terms of 'x' must be converted to the new variable 'u'. This transition is crucial as it ensures the definite integral is evaluated over the proper interval on the 'u' axis, which corresponds to the original integral's domain on the 'x' axis.

To find the new limits, we replace the original 'x' values with their corresponding 'u' values by substituting into the change of variables equation. For the lower limit of integration where \( x = 2 \), we compute \( u = 2^2 - 4 \), which yields \( u = 0 \). Similarly, for the upper limit where \( x = 4 \), we find \( u = 4^2 - 4 = 12 \).

Therefore, the new limits of integration for the variable 'u' are \( 0 \) and \( 12 \), respectively. Understanding these new limits is fundamental because they will define the new boundaries within which we'll integrate our function after successfully applying the change of variables.

Integration Techniques

To evaluate integrals effectively, various integration techniques are employed based on the form of the integrand. Along with change of variables, other methods include integration by parts, partial fraction decomposition, trigonometric substitution, and numerical integration.

Each technique has its own set of rules and applications. For example, the integration by parts is useful when the integrand is a product of two functions whose integration separately would be easier. Partial fraction decomposition is particularly suitable for rational functions where the denominator can be factored into linear or quadratic factors. Trigonometric substitution helps when integrating functions involving square roots of quadratic expressions that relate to trigonometric identities.

Knowing how to select the appropriate integration technique for a given problem is part of the art of calculus. It can make the difference between a quick, straightforward solution and a complex problem that becomes much harder to solve. In our exercise, the change of variables was crucial in simplifying the task, by transforming the given integral into one that's easier to work with in the context of the problem.