Problem 31 - Essential Calculus

Question

In Exercises 31-36, evaluate the integral using the following values. $$\int_{2}^{4} x^{3} d x=60, \quad \int_{2}^{4} x d x=6, \quad \int_{2}^{4} d x=2$$ $$ \int_{4}^{2} x d x $$

Step-by-Step Solution

Answer

-6

Step 1: Using the Property of Definite Integrals

The important thing to understand is that if we have an integral \(\int_{a}^{b} f(x) dx\), where \(b\) is greater than \(a\), and we reverse the limits of integration, we get \(\int_{b}^{a} f(x) dx = -\int_{a}^{b} f(x) dx\). This can be remembered as 'the reversal of the limits of integration changes the sign of the integral.' Applying this to \(\int_{4}^{2} x dx\), we find that it is equal to \(-\int_{2}^{4} x dx\).

Step 2: Substituting the Given Value

We are given \(\int_{2}^{4} x dx = 6\) from the problem. To find the value of \(\int_{4}^{2} x dx\) by substituting this into our previous equation, \(\int_{4}^{2} x dx = -\int_{2}^{4} x dx = -6\).

Key Concepts

Integration Properties

When working with integrals, especially definite integrals, knowing the properties of integration can simplify solving them. A key property is the ability to reverse the limits of integration. This property says that if you have an integral from point \( a \) to point \( b \), represented as \( \int_{a}^{b} f(x) \ dx \), you can swap \( a \) and \( b \) but you must multiply by \(-1\). Thus, \( \int_{b}^{a} f(x) \ dx = -\int_{a}^{b} f(x) \ dx \). This property is useful because sometimes reversing the limits can make use of known values or simplify calculations.
Another important property is the linearity of integration. If you have two functions, \( f(x) \) and \( g(x) \), their sum can be integrated separately: \( \int (f(x) + g(x)) \ dx = \int f(x) \ dx + \int g(x) \ dx \). This means we can evaluate integrals in manageable parts rather than tackling complex expressions all at once.
Even though these properties might seem basic, they serve as crucial tools in efficiently solving calculus problems. Make sure to harness them effectively when working on definite integrals.

Limits of Integration

Limits of integration determine the interval over which the function is being integrated. They are the values \( a \) and \( b \) in \( \int_{a}^{b} f(x) \ dx \). These values represent the start and end of the interval.
One significant aspect of limits is their impact on the sign of the integral. As demonstrated, reversing these limits inverts the integral's sign. This can be particularly handy when an integral's direction provides unnecessary complexity, allowing you to incorporate known quantities to simplify the task.
Besides affecting the sign, limits define the region where the area under the curve is calculated. When working with definite integrals, interpreting these correctly helps confirm you are evaluating the right portion of the function, ensuring the solution's accuracy. Comprehending how these limits affect the integral helps navigate through problems more clearly.

Calculus Exercises

Applying calculus to exercises often involves breaking down problems into simpler tasks by leveraging known integrals or applying properties we've discussed. In this exercise, knowing the property of reversing limits played a significant role.
Approaching a calculus exercise requires understanding which properties and rules simplify your work. Here, using the known value \( \int_{2}^{4} x \ dx = 6 \), combined with the properties of limits of integration, helped us find the integral \( \int_{4}^{2} x \ dx \). By simply reversing the sign, we get \( -6 \).
This technical maneuver underscores the importance of foundational calculus rules and concepts in more advanced problem-solving settings. Remember to always start with what you know and see if integrations properties or simple arithmetic can guide you to the solution efficiently.