Problem 27 - Calculus Single Variable

Question

In each of Exercises 25-30, use the method of cylindrical shells to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=x^{2}+1,\) above the \(x\) -axis, and between \(x=2\) and \(x=4\).

Step-by-Step Solution

Answer

The volume of the solid is \( 132\pi \).

Step 1: Understand the Problem

We're tasked with finding the volume of the solid formed by rotating a region, defined by the curve \( y = x^2 + 1 \), the x-axis, and the vertical lines \( x = 2 \) and \( x = 4 \), about the y-axis. This solid can be visualized as a set of cylindrical shells.

Step 2: Set Up the Integral for Volume

The volume \( V \) of the solid can be calculated using the formula for cylindrical shells: \[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \]where \( f(x) = x^2 + 1 \) is our function, and the limits of integration \( a = 2 \) and \( b = 4 \) are the given bounds of \( x \). Thus, the integral becomes:\[ V = 2\pi \int_{2}^{4} x(x^2 + 1) \, dx \]

Step 3: Simplify the Integrand

Expand the integrand by distributing \( x \) over \( x^2 + 1 \), resulting in:\[ x(x^2 + 1) = x^3 + x \]

Step 4: Evaluate the Integral

Calculate the integral:\[ \int_{2}^{4} (x^3 + x) \, dx = \left[ \frac{x^4}{4} + \frac{x^2}{2} \right]_{2}^{4} \]First, calculate the antiderivative:\[ \frac{x^4}{4} + \frac{x^2}{2} \]Evaluate from 2 to 4:\[ \left( \frac{4^4}{4} + \frac{4^2}{2} \right) - \left( \frac{2^4}{4} + \frac{2^2}{2} \right) \]

Step 5: Calculate the Definite Integral

Continuing the evaluation:\[ \left( \frac{256}{4} + \frac{16}{2} \right) - \left( \frac{16}{4} + \frac{4}{2} \right) \]This simplifies to:\[ (64 + 8) - (4 + 2) = 72 - 6 = 66 \]

Step 6: Calculate the Volume

Finally, multiply the result of the integral by \( 2\pi \) to get the volume:\[ V = 2\pi \times 66 = 132\pi \]

Step 7: Conclusion

Using the method of cylindrical shells, the volume of the solid generated by rotating the region \( y = x^2 + 1 \) from \( x = 2 \) to \( x = 4 \) around the y-axis is \( 132\pi \).

Key Concepts

Volume of Revolution

The volume of revolution is a fascinating concept in calculus that involves creating a 3D shape by rotating a 2D region about a specific axis. This process is especially useful for finding the volume of irregular shapes that cannot easily be measured with standard geometric formulas. In this context, we focus on understanding how to visualize and compute the volume when a region defined by a function is rotated around an axis, transforming it into a solid of revolution.
When using the cylindrical shells method, like in our exercise, we essentially slice the solid into thin cylindrical segments. Each cylindrical shell contributes to the total volume, and when combined, these shells accurately represent the entire solid. Understanding this concept is crucial for setting up the correct integral to calculate the volume.

Definite Integral

A definite integral is a powerful tool in calculus that allows us to evaluate the accumulation of quantities, such as areas under curves or volumes of solids. By defining the range of integration and an integrand (function that we're integrating), we can precisely quantify these areas or volumes.
In our example, the definite integral is used to sum up the contributions of each cylindrical shell to find the total volume of the solid of revolution.
The integral used is:
\[ V = 2\pi \int_{2}^{4} x(x^2 + 1) \, dx \]
The bounds here are critical—they define the interval over which we are summing, from \(x=2\) to \(x=4\), capturing the full extent of the generated solid. Through careful evaluation, this definite integral gives us the precise volume needed.

Solid of Revolution

A solid of revolution is formed when a region in a plane is revolved around a given line (axis). This rotation creates a 3D object whose volume can often be calculated with calculus. Such solids frequently arise in practical applications, making their understanding essential.
In our case, the solid of revolution is generated by rotating the region bounded by the function \(y = x^2 + 1\) from \(x = 2\) to \(x = 4\) about the y-axis.
Visualizing how this 2D region "revolves" around the axis helps in comprehending the shape of the solid. Each point in the region traces a circular pathway, forming a series of concentric shells, collectively comprising the solid.

Calculus Integration

Calculus integration is the mathematical process of finding the integral of a function, which essentially is about accumulating values to find areas, lengths, volumes, or other quantities. There are different techniques of integration, each useful in various circumstances.
For calculating the volume of a solid of revolution using cylindrical shells, the integration involves expanding and simplifying the integral to make solving more straightforward.
In our exercise, we expand the function by finding the antiderivative of the integrand:
\[ rac{x^4}{4} + rac{x^2}{2} \]
Then, we apply the Fundamental Theorem of Calculus by evaluating this at the upper and lower bounds, \(x=4\) and \(x=2\), to compute the definite integral. This meticulous step-by-step process is the essence of calculus integration, helping us solve complex real-world problems with precision.