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Magazine Chapter 7: Problem 6
In each of Exercises 1-6, use the method of disks to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(x\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=x^{1 / 3}\), above the \(x\) -axis, and between \(x=1\) and \(x=8\).
Short Answer
Expert verified
The volume of the solid is \( \frac{93\pi}{5} \).
Step by step solution
01
Understanding the Region of Revolution
First, identify the mathematical description of the region \( \mathcal{R} \). The curve given is \( y = x^{1/3} \), which is a function of \( x \). The region lies between \( x=1 \) and \( x=8 \), above the \( x \)-axis. Thus, \( \mathcal{R} \) is the area under \( y = x^{1/3} \) from \( x=1 \) to \( x=8 \). This region will be revolved around the \( x \)-axis to form the solid.
02
Setting Up the Integral
For the method of disks, the formula to find the volume \( V \) of the solid is given by \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \), where \( f(x) \) is the radius of the disk at a point \( x \). In this scenario, \( f(x) = x^{1/3} \). Therefore, the volume integral becomes \( V = \pi \int_{1}^{8} (x^{1/3})^2 \, dx \). Simplifying inside the integral gives \( V = \pi \int_{1}^{8} x^{2/3} \, dx \).
03
Calculating the Antiderivative
Next, compute the antiderivative of \( x^{2/3} \). Using the power rule for integration, the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) for any constant \( C \). So, \( \int x^{2/3} \, dx = \frac{x^{5/3}}{5/3} = \frac{3}{5} x^{5/3} \).
04
Evaluating the Definite Integral
Apply the limits of integration from 1 to 8 to the antiderivative: \[ V = \pi \left[ \frac{3}{5}x^{5/3} \right]_{1}^{8} = \pi \left( \frac{3}{5} imes 8^{5/3} - \frac{3}{5} \times 1^{5/3} \right) \].
05
Computing Numerical Values
Calculate \( 8^{5/3} \). This can be broken down as \( (8^{1/3})^5 = (2)^5 = 32 \). Thus, \( V = \pi \left( \frac{3}{5} \times 32 - \frac{3}{5} \times 1 \right) = \pi \left( \frac{96}{5} - \frac{3}{5} \right) = \pi \times \frac{93}{5} \).
06
Final Calculation
Simplify the expression to get \( V = \frac{93\pi}{5} \). This is the final volume of the solid formed by rotating the region around the \( x \)-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Method of Disks
The method of disks is used to find the volume of a solid of revolution, formed when a region is revolved around an axis. This technique involves imagining the solid being made up of an infinite number of thin, flat disks stacked together like a roll of coins. Each disk has a radius equal to the function value, and a thickness represented by a differential slice of the axis of rotation.
To apply this method, you must:
To apply this method, you must:
- Identify the function describing the edge of the solid, known as the radius of the disk.
- Determine the axis of rotation and set up the limits of integration.
- Using the integration formula for volume, where the integral of the area of a disk, \(\pi [f(x)]^2\ dx\), calculates the volume across the interval.
Definite Integrals
Definite integrals are key in calculating areas and volumes by integration, as they provide the total accumulated value over a specific interval. In the method of disks, definite integrals help compute the volume by summing up the values of the disks' areas across a defined region.
A definite integral \(\int_{a}^{b} f(x)\, dx\) calculates the net 'area' under a curve between two points \(a\) and \(b\). It's different from an indefinite integral as it gives a number, not a function.
A definite integral \(\int_{a}^{b} f(x)\, dx\) calculates the net 'area' under a curve between two points \(a\) and \(b\). It's different from an indefinite integral as it gives a number, not a function.
- The limits \(a\) and \(b\) are where the integration starts and ends.
- A definite integral accounts for the concept of adding up an infinite number of small quantities.
Antiderivatives
Calculating antiderivatives is a fundamental part of solving definite integrals. An antiderivative is a function whose derivative gives the original function. It's basically working backwards from a derivative to find the original function form.
When finding a volume using the method of disks, obtaining the antiderivative of the function describing the disk's area is necessary:
When finding a volume using the method of disks, obtaining the antiderivative of the function describing the disk's area is necessary:
- The definite integral results depend on the evaluated antiderivative.
- Solving an antiderivative requires understanding integration rules, such as the power rule.
Power Rule for Integration
The power rule for integration is a straightforward technique for finding the antiderivative of power functions. This rule is essential in solving many calculus problems where you need to integrate polynomial expressions.
The power rule states:
The power rule states:
- If you have a function \(x^n\), its antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the integration constant.
- To use it, increment the power by one and divide by the new power.
