Problem 137 - Calculus Volume 1

Question

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y = 0 around the x-axis. \(x=y^{3}-4 y^{2}, x=-1,\) and \(x=2\)

Step-by-Step Solution

Answer

Revolve using cylindrical shells with integral setup: \(2\pi \int y(y^3 - 4y^2 + 1) dy\), resolve between calculated limits.

Step 1: Identify Boundaries and Setup Integral for Volume

We need to find the volume of the solid formed by revolving the region around the x-axis. The boundaries given are between the curves \(x = y^3 - 4y^2\), \(x = -1\), and \(x = 2\). The axis of rotation is the x-axis.

Step 2: Express Radius and Height of Shells

For cylindrical shells, consider a shell at a distance \(y\) from the x-axis with height equal to the difference in \(x\) values given by \(x = y^3 - 4y^2\). Thus, the height of the shell is \((y^3 - 4y^2) - (-1) = y^3 - 4y^2 + 1\). Radius of each shell is \(y\).

Step 3: Determine Limits of Integration

We need to determine the y-values where \(y^3 - 4y^2 = -1\) and \(y^3 - 4y^2 = 2\). Solving \(y^3 - 4y^2 + 1 = 0\) and \(y^3 - 4y^2 - 2 = 0\) identifies the limits of integration. We'll solve these equations to find appropriate limits. (Assume or solve numerically if needed.)

Step 4: Set Up the Integral

The integral for the volume of the solid of revolution using cylindrical shells is \[V = 2\pi \int_{a}^{b} y \cdot (y^3 - 4y^2 + 1) \, dy\]where \(a\) and \(b\) are the y-values found in the previous step.

Step 5: Solve the Integral

Calculate the integral to find the volume:\[V = 2\pi \int_{a}^{b} (y^4 - 4y^3 + y) \, dy\]Evaluate this integral from \(a\) to \(b\).

Step 6: Evaluate the Final Result

Plug in the limits of integration, calculated from solving the equations in Step 3, into the antiderivative from Step 5, and find the volume. Multiply by \(2\pi\) to finalize the volume.

Key Concepts

Volume of Solids of Revolution

When we talk about the volume of solids of revolution, we mean the volume of a three-dimensional object created when a two-dimensional area is revolved around a line (the axis of rotation). In calculus, this is a common application when dealing with integrals. To find the volume of such solids, there are two primary methods:

• Disk Method: This method involves slicing the solid perpendicular to the axis of rotation, forming disks or washers. It is most effective when the object has a clear circular shape around the axis.
• Shell Method: Here, we imagine the solid is made up of thin, hollow cylindrical "shells." This is advantageous when the function or the region is easier to describe in a direction parallel to the axis of revolution.

For this specific exercise, the cylindrical shell method is more suitable because it handles revolving around the x-axis efficiently when the function is primarily in terms of y. Remember: Choosing the right method simplifies calculations and often makes setting up the integral easier.

Integration Techniques

Integration is the cornerstone of finding the volume of solids of revolution. Once we choose between the disk method and shell method, setting up the function into an integral form is critical.

• Setting Limits of Integration: The bounds of integration represent where the solid begins and ends along the axis of revolution. This is usually determined by the curves' intersections or provided constraints.
• Simplifying the Integrand: Once you identify the radius and height for the shell method, multiply these to get your integrand. Simplifying the integrand before integrating can save you from complex calculations later on.
• Antiderivative Calculation: Using known integration rules, your next task is to find the antiderivative of your integrand. This step often involves polynomial, trigonometric, or other functions depending on the shape being rotated.

Keep in mind: Always check if simplifications or algebraic manipulations can ease your integration workload, and ensure your integral represents the actual physical context you're analyzing.

Cylindrical Shells

The cylindrical shell method is an elegant technique for calculating volumes when revolving curves. Think of each shell as a hollow, thin cylinder with its axis around the line of revolution.

• Components of a Shell: Each shell has a height, determined from the function's value, a radius, which is the distance from the axis of rotation, and a thickness represented by a small change in the variable of integration ( \( dy \) when integrating with respect to y).
• Formula Explanation: The formula \( V = \int 2\pi \cdot \text{{radius}} \cdot \text{{height}} \cdot \text{{thickness}} \) guides the setup of the integral. Here, the \(2\pi\) accounts for the circular movement around the axis and combined with your function, gives the volume.
• Application Benefits: The shell method is particularly helpful if the region can be easily described in parallel layers to the axis of rotation or when dealing with functions defined by more than one variable, like x in terms of y.

In our current exercise, setting up the shells correctly ensures we fully capture the volume generated by revolving the given region. Carefully determining each shell's components makes the shell method both powerful and precise.