Problem 134 - 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. $$x=\frac{1}{1+y^{2}}, y=1, \text { and } y=4$$

Step-by-Step Solution

Answer

The volume is \(\pi \ln(8.5)\).

Step 1: Visualize the Problem

Visualize the region described by the function and boundaries. We have the curve \(x = \frac{1}{1+y^2}\), bounded by \(y = 1\) and \(y = 4\). We are rotating this region around the \(x\)-axis.

Step 2: Understand the Shell Method Formula

The volume \(V\) using the shell method when rotating around the \(x\)-axis is given by the integral formula: \[ V = 2\pi \int_{a}^{b} y \, (x(y)) \ dy \] where \(x(y)\) is a function of \(y\), and \(a\) and \(b\) are the bounds of \(y\).

Step 3: Set Up the Integral

Substitute \(x(y) = \frac{1}{1+y^2}\), \(a = 1\), and \(b = 4\) into the formula: \[ V = 2\pi \int_{1}^{4} y \left(\frac{1}{1+y^2}\right) \, dy \]This simplifies to:\[ V = 2\pi \int_{1}^{4} \frac{y}{1+y^2} \, dy \]

Step 4: Solve the Integral

To solve \(\int \frac{y}{1+y^2} \, dy\), use the substitution method. Let \( u = 1 + y^2 \), then \( du = 2y \, dy \). Therefore, \( y \, dy = \frac{1}{2} du \). Substituting gives:\[ \frac{1}{2} \int \frac{1}{u} \, du \]which simplifies to \(\frac{1}{2} \ln|u| + C\). Re-substitute \(u\) back:\[ \frac{1}{2} \ln|1+y^2| + C \]

Step 5: Evaluate the Integral from 1 to 4

Now evaluate the antiderivative: \[ V = 2\pi \left[ \frac{1}{2} \ln|1+y^2| \right]_{1}^{4} \]This becomes:\[ V = \pi \left( \ln(1+16) - \ln(2) \right) \]Simplify further:\[ V = \pi \left( \ln(17) - \ln(2) \right) \]Using the property \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), we have:\[ V = \pi \ln\left(\frac{17}{2}\right) \]

Step 6: Final Calculation

Perform the final calculation:\[ V = \pi \ln\left(8.5\right) \]Thus, the volume of the solid generated by rotating the given region around the \(x\)-axis is \(\pi \ln(8.5)\).

Key Concepts

Shell Method

The Shell Method is a wonderful tool for calculating the volume of a solid formed by rotating a region around an axis. Imagine spinning a cylindrical shell around an axis, creating a hollow cylinder. That's the basic idea behind the Shell Method. To use it effectively, you consider small vertical slices or shells of the region you want to rotate.

• The volume of one such shell is given by its circumference times its height and thickness.
• This is summed up over the interval to get the total volume.
• Mathematically, this is expressed as the integral of the shells over a given interval.

In the original exercise, we rotated the region around the x-axis using shells. It's crucial to express everything in terms of the y-variable since we are rotating about the horizontal axis. The integral setup reflects the sum of shells' volumes, showcasing their utility in solving complex geometric problems.

Volume of Revolution

The Volume of Revolution allows us to find the space occupied by a 3D object when a 2D shape rotates around an axis. Consider how a flat region, when spun, forms a three-dimensional object.

• There are two popular methods: the Shell and Disk/Washer methods.
• In both cases, identifying the region and the axis of rotation is key.
• Choosing the right method simplifies the calculation process.

In our exercise, the focus was on using the Shell Method to find the volume of a shape created when the region between the curve \( x = \frac{1}{1+y^2} \) and \( y = 0 \) is rotated around the x-axis.The resulting volume involves calculating an integral that reflects the shape's geometry as it revolves, revealing the power of calculus in visualizing and quantifying shapes.

Definite Integration

Definite Integration is your ultimate tool for summing up things over an interval. It's like adding up infinite slices to find the total. When calculating volumes, it helps in accumulating little bits of volume from each shell or disk.

• You evaluate it between two bounds, giving a finite result.
• The process often involves antiderivatives, substitution, or other calculus techniques.
• It's perfect for calculating areas, lengths, and volumes.

In our specific problem, we used definite integration to evaluate the volume. The integral \( \int_{1}^{4} \frac{y}{1+y^2} \, dy \) had clear bounds and was solved using substitution. Here, it allowed us to compute the precise volume of the rotated shape. By understanding definite integration deeply, you unlock the ability to analyze and solve real-world shape and space problems effectively.