Problem 33 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The integral \( \int_{2}^{4} 4x dx \) equals 24.

Step 1: Spot the constant in the integral

In the integral \( \int_{2}^{4} 4x dx \), the number 4 is a constant which is being multiplied with the function \(x\). In the integral calculus, constants can be factored out of the integral.

Step 2: Factor out the constant

The constant 4 can be factored out of the integral: \( \int_{2}^{4} 4x dx = 4 \int_{2}^{4} x dx \). This makes the integral easier to solve because we already know the value of \( \int_{2}^{4} x dx \) from the given data.

Step 3: Substitute the value of the integral

Now, substitute the given value \( \int_{2}^{4} x dx = 6 \) into the integral equation: \( 4 \int_{2}^{4} x dx = 4 \cdot 6 \).

Step 4: Solve for the final value

Finally, just simplify \( 4 \cdot 6 \) to get 24. This is the value of \( \int_{2}^{4} 4x dx \).

Key Concepts

Definite Integrals

Understanding definite integrals is a foundational skill in calculus. A definite integral represents the accumulation of quantities, such as areas under a curve, over a specific interval. In symbolic terms, if you have an integral of the form \( \int_{a}^{b} f(x) dx \), it represents the sum of the values of \( f(x) \) from \( x=a \) to \( x=b \).

In our exercise, \( \int_{2}^{4} x^{3} dx=60 \) represents the area under the curve of \( x^3 \) between \( x=2 \) and \( x=4 \). Definite integrals have properties that make them easier to evaluate, such as linearity and the ability to add or subtract over separate intervals. These properties, along with known values of certain integrals, help streamline the process of calculation and assist in understanding the behavior of functions over an interval.

Constant Factor Rule in Integrals

The constant factor rule is a simplification technique used in integral calculus. It states that if you have a constant \( k \) multiplied by a function \( f(x) \) inside an integral, you can take the constant out of the integral. So, the integral \( \int k \cdot f(x) dx \) becomes \( k \int f(x) dx \).

In our textbook exercise, this rule is applied to \( \int_{2}^{4} 4x dx \). The constant 4 is pulled out of the integral, simplifying the problem to \( 4 \int_{2}^{4} x dx \). With the constant factor rule, we effectively separate the complexity of constant multiplication from the process of integration, which can often make integral evaluation significantly simpler. This rule is greatly advantageous when dealing with complicated functions that, when stripped of their constants, become recognizably simpler to deal with.

Integration Properties

Integration has several properties that are pivotal for simplifying and solving integration problems. Some of these properties include the additivity of integrals, the zero factor property, and as mentioned earlier, the constant factor rule.

Additivity allows you to break down a complex integral into simpler, more manageable parts, and then add the result of each part. The zero factor property says that the integral of zero over any range is zero. These properties play a crucial role in solving integrals efficiently and are especially helpful when applying techniques of integration to find areas, volumes, and other physical quantities.

Understanding and applying these properties correctly can convert a seemingly daunting integral into a straightforward calculation. In our textbook solution, after applying the constant factor rule, we use another property of integrals—the ability to substitute known integral values to reach a final answer without performing the full integration process.