Problem 22 - Thomas Calculus

Question

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\)-axis. \(y=x^{3}, \quad y=0, \quad x=2\)

Step-by-Step Solution

Answer

The volume of the solid is \( \frac{128\pi}{7} \).

Step 1: Understand the Problem

We need to calculate the volume of the solid formed when the region between the curve \( y = x^3 \) and the line \( y = 0 \) from \( x = 0 \) to \( x = 2 \) is rotated around the \( x \)-axis.

Step 2: Apply Disk Method Formula

The volume \( V \) of the solid of revolution can be found using the disk method formula: \[ V = \, \int_{a}^{b} \pi [f(x)]^2 \, dx \]Here, the function \( f(x) = x^3 \), and the limits of integration \( a = 0 \) and \( b = 2 \).

Step 3: Substitute and Integrate

Substitute \( f(x) = x^3 \) into the formula, then integrate:\[ V = \int_{0}^{2} \pi (x^3)^2 \, dx \] This simplifies to:\[ V = \pi \int_{0}^{2} x^6 \, dx \]

Step 4: Compute the Integral

Calculate the integral:1. Find the antiderivative: \[ \int x^6 \, dx = \frac{x^7}{7} \]2. Evaluate from 0 to 2: \[ V = \pi \left[ \frac{x^7}{7} \right]_{0}^{2} \]

Step 5: Evaluate the Limits of the Integral

Substitute the limits into the antiderivative:\[ V = \pi \left[ \frac{2^7}{7} - \frac{0^7}{7} \right] = \pi \left[ \frac{128}{7} - 0 \right] \]

Step 6: Final Calculation

Calculate the numerical value:\[ V = \pi \times \frac{128}{7} = \frac{128\pi}{7} \]

Key Concepts

Disk Method

The **disk method** is a technique used to find the volume of a solid of revolution. This happens when a region in the plane is revolved around a line, such as the x-axis or y-axis. The resulting solid's volume can be found by approximating the solid with a series of thin, flat disks. To use the disk method effectively, follow these steps:

• Identify the function that represents the boundary of the region you rotate.
• Determine the axis of rotation. In our problem, it's the x-axis.
• Decide on the limits of integration, which are the bounds within which the region exists. For the given exercise, the bounds are from 0 to 2.

The formula to calculate the volume is:\[ V = \, \int_{a}^{b} \pi [f(x)]^2 \, dx \]This formula calculates the sum of the volumes of all the disks, giving the total solid volume.

Definite Integral

A **definite integral** is a fundamental concept in calculus that helps calculate the exact area under a curve between two points, or in this case, the exact volume of a solid of revolution. When you compute the definite integral of a function over a specific interval, you get a number, which represents this "area" or "volume."For the problem at hand, the definite integral formula is:\[ V = \pi \int_{0}^{2} x^6 \, dx \]Here, the bounds of the integral (0 and 2) let us know the region of interest. This integral provides the means to compute the precise volume when revolved around the x-axis by considering the "slices" represented by the disks.

Antiderivative

The **antiderivative**, or indefinite integral, represents the reverse process of differentiation. It provides a function whose derivative is the given function. To use definite integrals to find area or volume, one finds the antiderivative first.For our integration step with \( x^6 \):

• The antiderivative of \( x^6 \) is computed as \( \int x^6 \, dx = \frac{x^7}{7} \)
• This expression gives us a new function to evaluate over the specific interval.

Using this antiderivative, you then substitute in the boundaries of the region to find the solid's volume, leading to the numerical evaluation.

Solid of Revolution

A **solid of revolution** is created when a plane area is revolved around a line (the axis) to create a three-dimensional shape. This concept is often applied using the disk, washer, or shell methods in calculus.In the exercise, revolving the area bounded by \( y = x^3 \) and \( y = 0 \) around the x-axis produces a solid of revolution. This specific solid has rotational symmetry around the x-axis.Such solids can vary widely in shape based on the curve revolved and the axis chosen, but calculating their volumes typically involves methods like the disk method discussed earlier. Understanding these concepts allows you to tackle a variety of calculus problems involving three-dimensional shapes formed by rotation.