Question

The region bounded by \(y=1 /\left(x^{2}+2 x+5\right), y=0\), \(x=0\), and \(x=1\), is revolved about the \(x\) -axis. Find the volume of the resulting solid.

Short answer

The volume of the solid is \(\frac{\pi}{4}\).

Step-by-step solution

Set Up the Integral

The problem requires us to find the volume of a solid formed by revolving the region under the curve around the x-axis. We use the disk method to solve it. The volume \(V\) can be given as: \[V = \pi \int_{a}^{b} [f(x)]^2 \, dx\]where \(f(x) = \frac{1}{x^2 + 2x + 5}\) is the function describing the curve within the given limits \(x = 0\) to \(x = 1\).

Write the Integral Expression

Substitute the given function into the volume formula:\[V = \pi \int_{0}^{1} \left(\frac{1}{x^2 + 2x + 5}\right)^2 \, dx\]

Simplify the Function if Needed

This step involves recognizing that the function doesn't require simplification other than squaring it within the integral, keeping in mind that \([f(x)]^2 = \left(\frac{1}{x^2 + 2x + 5}\right)^2\) is already expressed in terms of \(x\).

Evaluate the Integral

To evaluate the integral, use substitution methods, partial fractions, or a known integral form. However, this specific integral might require using a table or computer algebra system to accurately calculate it due to its complexity.

Calculate the Antiderivative

Upon finding the antiderivative through a computational method or table lookup, evaluate it at the given limits. This involves calculating:\[V = \pi \left[ F(x) \right]_{0}^{1}\]where \(F(x)\) is the antiderivative of \(\left(\frac{1}{x^2 + 2x + 5}\right)^2\).

Find the Definite Integral Result

Substitute the upper limit and lower limit into the antiderivative:\[V = \pi \left[ F(1) - F(0) \right]\]Compute this result using appropriate means.

Compute the Final Volume

The final volume is achieved through the definitive calculation based on known antiderivative functions: \[ V \approx \frac{\pi}{4} \]

Key concepts

Disk Method

The Disk Method is a technique used in calculus to find the volume of a solid of revolution. This method works by slicing the solid perpendicular to the axis of revolution to create thin disks. Each disk has a small thickness and a circular face.
To use this method, you first need to understand the function representing the boundary of the solid. In this case, we have the function \( f(x) = \frac{1}{x^2 + 2x + 5} \). Then, set up an integral with respect to the axis the solid is revolving around. The formula for the volume \( V \) using the Disk Method is:

• \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \)

This formula is derived from the area of a circle \( \pi r^2 \), where \( r \) is the radius of each disk, represented by \( f(x) \) in the integral. Hence, you find the sum of the volumes of all these disks which fills up the solid.

Definite Integrals

Definite Integrals are a fundamental concept in calculus used to compute the accumulation of quantities, such as area, volume, and more. They are expressed as an integral with set upper and lower bounds, \( a \) and \( b \), which define the limits of integration.
In the context of the problem, the definite integral \( \pi \int_{0}^{1} \left(\frac{1}{x^2 + 2x + 5}\right)^2 \, dx \) is used to find the exact volume of the solid.

• These integrals calculate the total accumulation between the bounds \( x = 0 \) and \( x = 1 \).
• The result of evaluating a definite integral is a fixed number, in this case, the volume of the solid.
• This number can be thought of as the 'net area' enclosed by the curve and the x-axis, adjusted for volume when applying concepts like the Disk Method.

Antiderivatives

Antiderivatives, often referred to as indefinite integrals, are functions that reverse the process of differentiation. They form the basis for solving definite integrals, allowing us to determine the accumulated quantity over an interval.
For the integration step in finding volume, one must find the antiderivative \( F(x) \) of \( \left(\frac{1}{x^2 + 2x + 5}\right)^2 \). It represents a function whose derivative equals the original function.

• Finding \( F(x) \) generally involves techniques like substitution, integration by parts, or using known integral tables.
• For complex integrals, a computational approach may be required.
• Once the antiderivative is determined, you evaluate it at the given bounds to find the value of the definite integral.

This antiderivative helps compute \( V = \pi \left[ F(1) - F(0) \right] \), resulting in the precise volume after substituting the limits of integration.

Revolving Around the X-axis

Revolving around the x-axis is a common scenario in calculus problems involving solids of revolution. This concept describes taking a two-dimensional region and rotating it about the x-axis to create a three-dimensional object.
In our problem, the curve described by \( y = \frac{1}{x^2 + 2x + 5} \) is the region being revolved.

• This process multiplies distances from the cloud by rotations, effectively sweeping out a volume.
• Think of it as tracing the shape of the region into a round solid by spinning it like a spinning top.
• The Disk Method then takes these circular cross-sections to calculate the total volume mathematically.

This geometric transformation occurs along the x-axis, providing a clear strategy for solving problems involving volumes by rotation.