Problem 25 - Applied Calculus

Question

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=8 x^{3} \text { from } x=1 \text { to } x=3 $$

Step-by-Step Solution

Answer

The area is 160 square units.

Step 1: Identify the Function and Limits

We are given the function \( f(x) = 8x^3 \). We need to find the area under this curve from \( x = 1 \) to \( x = 3 \). This means our limits of integration are from \( x = 1 \) to \( x = 3 \).

Step 2: Set Up the Definite Integral

Since we want to find the area under the curve \( f(x) \) from \( x=1 \) to \( x=3 \), we set up the definite integral:\[\int_{1}^{3} 8x^3 \, dx\]

Step 3: Integrate the Function

To integrate \( 8x^3 \), we use the power rule for integration. The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). Thus, the indefinite integral of \( 8x^3 \) is:\[\int 8x^3 \, dx = 8 \cdot \frac{x^{4}}{4} = 2x^4\]

Step 4: Evaluate the Definite Integral

Now we need to evaluate the definite integral from \( x = 1 \) to \( x = 3 \). This requires calculating:\[\left[ 2x^4 \right]_{1}^{3} = 2(3)^4 - 2(1)^4\]

Step 5: Calculate the Values

Calculate \( 2(3)^4 \) and \( 2(1)^4 \):\[ 2(3)^4 = 2 \times 81 = 162\]\[ 2(1)^4 = 2 \times 1 = 2\]

Step 6: Find the Difference

Subtract the lower limit value from the upper limit value:\[ 162 - 2 = 160\]

Step 7: Conclusion and Sketch

The area under the curve \( f(x) = 8x^3 \) from \( x=1 \) to \( x=3 \) is 160 square units. For the sketch, plot the curve of \( f(x) = 8x^3 \) and shade the region between \( x = 1 \) and \( x = 3 \).

Key Concepts

Area Under a Curve

The area under a curve represents a significant concept in calculus, especially when dealing with functions and their integrals. When we talk about the "area under a curve," we are referring to the region between the graph of a function and the x-axis over a specified interval. This region can be pictured as the accumulation of tiny strips of space between the curve and the horizontal axis.

In the given exercise, the area under the curve of the function \( f(x) = 8x^3 \) from \( x = 1 \) to \( x = 3 \) represents a specific quantity. The definite integral is used to calculate this area. The integral sums up an infinite number of infinitesimally small areas to give us the total area between these two points. This total area can be interpreted as the net accumulated value of the function over that interval.

Understanding this concept allows us to evaluate how functions behave within given boundaries and offers a way to quantify their overall impact over a given region in terms of area.

Integration Techniques

Integration techniques refer to various methods used to find the integral of a function. For the function \( f(x) = 8x^3 \), a specific choice of technique depends on the structure of the function. Here, the power rule is most suited as it directly applies to functions of the form \( x^n \).

Some of the frequently used integration techniques include:

• Power rule: Used for integrating polynomial functions and is applicable to our exercise example where we integrate \( 8x^3 \).
• Substitution: Useful for more complex functions involving compositions, allowing for simplification before integration.
• Integration by parts: Employed when dealing with functions that can be expressed as a product of two simpler functions.

Each technique provides a strategic way to tackle integrals, facilitating more efficient and manageable computations. Knowing when and how to apply these methods is key in solving definite integrals like the one in this exercise.

Power Rule for Integration

The power rule for integration is a fundamental technique used to integrate polynomial functions. It is a direct and straightforward method suitable for functions in the form of \( x^n \), where \( n \) is a real number.

Applying the power rule, the integral of \( x^n \) is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \( C \) is the integration constant.

In the case of a definite integral, the power rule helps in finding the exact area under the curve by evaluating:\[\int_{a}^{b} x^n \, dx = \left[ \frac{x^{n+1}}{n+1} \right]_{a}^{b}\]For the function \( f(x) = 8x^3 \), applying the power rule helps calculate the integral \( 2x^4 \) as shown in the original step-by-step solution.

Understanding this rule significantly simplifies the process of finding definite integrals, making it a cornerstone in calculus for both study and application.