Problem 44 - Precalculus with Limits

Question

In Exercises 37-48, use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. $$ f(x) = 64 - x^3 $$ Interval \( [1, 4] \)

Step-by-Step Solution

Answer

The area of the region between the graph of the function \(f(x) = 64 - x^3\) and the x-axis over the interval [1, 4] is \(128.25\).

Step 1: Set up the integral

First set up the integral based on the given function and interval. The integral of \(f(x) = 64 - x^3\) from 1 to 4 is given by:\[\int_{1}^{4} (64 - x^3)\,dx\]

Step 2: Evaluate the integral

Now, evaluate the integral. When integrating, \(64\) becomes \(64x\), and \(x^3\) becomes \(\frac{1}{4}x^4\). So, the integral simplifies as:\[\int_{1}^{4} (64 - x^3)\,dx = [64x - \frac{1}{4}x^4]_{1}^{4}\]Next, plug in the bounds of the interval (1 and 4) into \(x\):\[= [64*4 -\frac{1}{4}*4^4] - [64*1 - \frac{1}{4}*1^4]\]

Step 3: Simplify the expression

Now, simplify the expression to find the area under the curve. \[= [256 - 64] - [64 - 0.25]\]After simplifying, the solution to the problem becomes:\[= 192 - 63.75 = 128.25\]

Key Concepts

Limit Process

The limit process is fundamental in integral calculus, allowing us to find the area under a curve. Essentially, it involves approximating the area by dividing it into thin rectangular strips of equal width, known as subintervals. As the number of these subintervals increases, the width of each strip gets smaller, leading to a more accurate approximation of the area.

When we express this idea mathematically, we establish the concept of a definite integral. The smaller the width of the strips, the closer their combined area approximates the true area under the curve. Hence, the limit process refers to taking the limit of this approximation as the width of the rectangles approaches zero.

• The goal is to refine our approximation of the area.
• With more strips, we achieve a more accurate result.
• This process lays the groundwork for calculating definite integrals.

Area Under a Curve

Finding the area under a curve is a common application of integral calculus. The function in this exercise, \( f(x) = 64 - x^3 \), defines a region between the curve and the x-axis over the interval from 1 to 4. This technique is essential in many fields, including physics and economics, to determine various quantitative measures like distance, work, or cost.

The area under a curve is typically calculated using a definite integral. By integrating the function over the specified interval, we effectively sum up all the small areas of the rectangular strips we discussed earlier. In this scenario, it allows us to quantify the space between the graph of \( f(x) \) and the x-axis, providing a numerical value representing the total area.

Definite Integral

A definite integral represents the limit of a sum of areas of rectangles under a curve as their width approaches zero. In the given exercise, we evaluate the integral \( \int_{1}^{4} (64 - x^3) \,dx \) to determine the area under the curve of the polynomial function. It provides a clear method for calculating areas by considering both the function values and the interval.

The process involves evaluating the antiderivative at the upper and lower bounds of the interval and finding their difference. For this exercise, the antiderivative of \( 64 - x^3 \) is \( 64x - \frac{1}{4}x^4 \). Plugging in the values of the bounds gives us the exact area, which simplifies to 128.25.

• A definite integral is always evaluated between given limits.
• It involves finding the antiderivative and then calculating the difference between its values at the bounds.
• This contrasts with an indefinite integral, which is not evaluated over a specific interval.

Polynomial Functions

Polynomial functions are algebraic expressions involving variables raised to non-negative integer powers. The function in this problem, \( f(x) = 64 - x^3 \), is a polynomial function where the highest power is cube (3rd degree). These functions are widely studied in calculus because they are smooth and continuous, making them amenable to various analytical techniques, including differentiation and integration.

• They consist of terms like \(x^0, x^1, x^2, \ldots \).
• The degree of the polynomial is decided by the highest power of the variable.
• Polynomial functions form the backbone of key calculus operations.

Integrating polynomial functions involves applying basic rules of integration, like increasing the power by one and dividing by the new power, to each term separately. For higher-degree polynomials, this becomes crucial for finding areas and solving real-world problems where polynomial behaviors are observed.