Problem 118 - Calculus Volume 1

Question

For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. [T] Under the curve of \(y=2 x^{3}, x=0,\) and \(x=2\) rotated around the \(y\) -axis.

Step-by-Step Solution

Answer

The volume is \( \frac{128\pi}{5} \). Both methods confirm this result.

Step 1: Graph the Functions

We begin by sketching the graph of the function \( y = 2x^3 \) between \( x = 0 \) and \( x = 2 \). This curve passes through the points \( (0,0) \), \( (1, 2) \), and \( (2, 16) \). We will rotate the area under this curve around the \( y \)-axis.

Step 2: Define the Shell Method Formula

The method of cylindrical shells involves integrating the volume of cylindrical shells. The formula for the volume \( V \) using the shell method when rotating about the \( y \)-axis is given by \[ V = 2\pi \int_{a}^{b} x (f(x) - g(x)) \, dx \] where \( f(x) \) is the outer function and \( g(x) \) is the inner function. Here, \( a = 0 \) and \( b = 2 \).

Step 3: Apply Shell Method for Rotation

For our curve \( y = 2x^3 \), the inner function \( g(x) = 0 \). Substituting into the shell method formula, we have: \[ V = 2\pi \int_{0}^{2} x \cdot 2x^3 \, dx = 2\pi \int_{0}^{2} 2x^4 \, dx. \] Simplifying, \[ V = 4\pi \int_{0}^{2} x^4 \, dx. \] We find the integral: \[ V = 4\pi \left[ \frac{x^5}{5} \right]_{0}^{2}. \] Evaluating, \[ V = 4\pi \left( \frac{32}{5} \right) = \frac{128\pi}{5}. \]

Step 4: Define the Washer Method Formula

The volume \( V \) using the washer method involves integrating the cross-sectional area of washers. The formula is given by \[ V = \pi \int_{a}^{b} (R^2 - r^2) \, dy \] where \( R \) is the outer radius function and \( r \) is the inner radius function. We need to express \( x \) as a function of \( y \).

Step 5: Express x as a Function of y

We have \( y = 2x^3 \). Solving for \( x \), we get \( x = \left(\frac{y}{2}\right)^{1/3} \). Therefore, the limits for \( y \) are from \( y = 0 \) to \( y = 16 \).

Step 6: Apply Washer Method for Rotation

The outer radius \( R(y) = \left(\frac{y}{2}\right)^{1/3} \) and the inner radius \( r(y) = 0 \). Substituting into the washer method formula,\[ V = \pi \int_{0}^{16} \left( \left(\frac{y}{2}\right)^{2/3} - 0 \right) \, dy \] This simplifies to:\[ V = \pi \int_{0}^{16} \left(\frac{y}{2}\right)^{2/3} \, dy. \] Further simplifying:\[ V = \pi \left( \frac{1}{2} \right)^{2/3} \int_{0}^{16} y^{2/3} \, dy. \]

Step 7: Evaluate the Integral for Washer Method

The integral \( \int_{0}^{16} y^{2/3} \, dy \) is solved as follows: \[ V = \pi \left( \frac{1}{2} \right)^{2/3} \left[ \frac{y^{5/3}}{5/3} \right]_{0}^{16}. \] This evaluates to: \[ V = \pi \left( \frac{1}{2} \right)^{2/3} \left( \frac{3}{5} \cdot 64 \right) = \frac{128\pi}{5}. \]

Step 8: Comparison of Results from Both Methods

Both methods – the shell method and the washer method – yield the same volume result: \( V = \frac{128\pi}{5} \). This confirms the accuracy of our calculations.

Key Concepts

Shell Method

The shell method is a technique used to find the volume of a solid of revolution, specifically when a region is rotated about an axis.Using the cylindrical shell approach, we can visualize the volume as being made up of several hollow cylinders or 'shells'. For a rotation around the y-axis, the volume of each shell is determined by its circumference, height, and thickness. The formula for the shell method, \[ V = 2\pi \int_{a}^{b} x (f(x) - g(x)) \, dx \] helps integrate these elements from the lower limit \( a \) to the upper limit \( b \). - The circumference is given by \( 2\pi x \), where \( x \) is the distance from the y-axis.- The height is determined by the function \( f(x) - g(x) \), representing the outer and inner shell functions.- Finally, the thickness is a small change in \( x \) or \( dx \). This method simplifies finding volumes when rotating complex curves around a vertical axis, avoiding the need for calculating inverse functions which can be cumbersome.

Washer Method

The washer method is another powerful tool in calculus, particularly for calculating volumes of revolution around an axis that is horizontal.With this method, we conceive the solid as a series of "washers". These washers are essentially disks with holes, imagined to stack along the axis. The formula for this method, \[ V = \pi \int_{a}^{b} (R^2 - r^2) \, dy \]allows us to compute the volume by subtracting the volume of the inner disk (hole) from the outer disk.- The outer radius \( R(y) \) is defined by the furthest curve from the axis of rotation.- The inner radius \( r(y) \) is given by the nearest curve.- The thickness is a small change in \( y \) or \( dy \). This method is particularly useful when the limits of integration are easily set in terms of \( y \) and when dealing with revolutions around the x-axis. Understanding it helps with visualizing and breaking down more intricate shapes into simpler components for volume calculation.

Calculus Integration

Integration is a fundamental concept in calculus, particularly essential when solving problems involving areas, volumes, and other quantities. In the context of finding volumes of revolution, integration helps sum up infinitesimally small elements over a continuous range. Both the shell and washer methods apply integration to measure the volume formed when a planar region is revolved around an axis. - **Shell Method**: Integrates using vertical elements to account for radial distance. - **Washer Method**: Sums horizontal slices to exclude inner hollow parts. These methods refine our approach to handling otherwise complicated volumes through straightforward integration. Mastering these techniques provides a deeper understanding of how shapes are altered through rotation, aiding in visualizing three-dimensional geometry from two-dimensional functions.

Cylindrical Shells

Cylindrical shells contain a critical visualization approach for tackling volume of revolution problems using the shell method.Imagine peeling an orange such that the peel represents a thin cylindrical shell. As we revolve a thin slice of a function around an axis, this conceptually forms similar shells. - A cylindrical shell corresponds to a small, thin cylinder in our volume calculation.- The height of this shell is given by the difference in function values \( f(x) - g(x) \).- The average radius is \( x \, ( \text{or } y \text{, depending on the axis}) \), making the shell's circumference \( 2\pi x \). These elements are integrated over the desired interval. By considering the collective volume of all such shells, the total volume of the solid revolution is realized comprehensively.Cylindrical shells streamline the integration process needed for volume calculations, fostering an easier understanding of how rotating a region generates a solid in three-dimensional space.