Problem 30 - Calculus and its applications

Question

Find the volume generated by rotating the area bounded by the graphs of each set of equations around the \(y\) -axis. $$ y=\frac{1}{x^{2}+1}, x=1, x=7 $$

Step-by-Step Solution

Answer

Volume is approximately \(10.118\pi\).

Step 1: Understand the Problem

We need to find the volume of the solid formed by rotating the region under the curve \(y = \frac{1}{x^2 + 1}\) from \(x = 1\) to \(x = 7\) around the \(y\)-axis. This involves using the method of cylindrical shells.

Step 2: Set Up the Cylindrical Shell Formula

The volume \( V \) of a solid of revolution using the cylindrical shell method is given by the integral \[ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \] where \(f(x)\) is the given function, and \(a\) and \(b\) are the bounds of \(x\). Here \(f(x) = \frac{1}{x^2+1}\), \(a = 1\), and \(b = 7\).

Step 3: Write the Integral Expression

Using the formula for cylindrical shells, the integral becomes: \[ V = 2\pi \int_{1}^{7} x \cdot \frac{1}{x^2+1} \, dx \]

Step 4: Simplify the Integral

Perform the simplification to make the integral easier to solve. The integral simplifies to: \[ V = 2\pi \int_{1}^{7} \frac{x}{x^2+1} \, dx \]

Step 5: Solve the Integral

To solve the integral \(\int \frac{x}{x^2+1} \, dx\), use the substitution method. Let \( u = x^2 + 1 \), then \( du = 2x \, dx \) or \( x \, dx = \frac{1}{2} du \). Substituting these into the integral gives: \[ V = 2\pi \int \frac{1}{2} \cdot \frac{1}{u} \, du = \pi \int \frac{1}{u} \, du \] The antiderivative is \(\ln|u|\), so: \[ V = \pi \ln|x^2+1| \] Evaluating this from \(x = 1\) to \(x = 7\) gives: \[ V = \pi [\ln(7^2+1) - \ln(1^2+1)] = \pi \ln\frac{50}{2} = \pi \ln 25 \]

Step 6: Compute the Final Volume

Plug the values into the expression to obtain the numerical result: \[ V = \pi \ln 25 \] Calculate the numerical approximation if needed using \(\ln 25 \approx 3.21888\), so \( V \approx \pi \times 3.21888 \approx 10.1175 \pi \).

Key Concepts

Volume of Revolution

Calculating the volume of a solid of revolution is a captivating part of calculus that allows us to find the volume of a 3D object formed by rotating a 2D area around an axis. In this exercise, the region under the curve defined by the equation \( y = \frac{1}{x^2+1} \) is rotated around the \( y \)-axis.
The volume calculation method used here is the cylindrical shells method. This method is particularly useful when rotating around the \( y \)-axis, as it involves summing up an infinite number of thin cylindrical shells that make up the entire volume.

• Think of peeling an onion layer by layer; each layer represents a cylindrical shell in this context.
• The beauty of this method lies in its ability to tackle more complex curves where traditional disc or washer methods would require extensive computation.

By using cylindrical shells, we effectively construct the volume by integrating across the width of the region while considering its distance from the axis of rotation, its height, and its circumference.

Integration Techniques

Integration is the process of finding the area under a curve, and in this exercise, it's essential for calculating the volume of revolution. The integral expression we derive is:
\[ V = 2\pi \int_{1}^{7} \frac{x}{x^2+1} \, dx \]
In calculus, integrating such expressions where there's a fraction involved involving a polynomial can be simplified using certain techniques

• Cylindrical shell volume formulas are especially useful as they incorporate circumference and height into the integral.
• This particular form suggests the usage of logarithmic integration, as we have \( \frac{x}{x^2+1} \) which structurally hints towards a solution involving natural logarithms.

Here, detecting these kinds of integral forms makes it significantly easier to solve, which leads us onto our next topic: substitution method.

Substitution Method

The substitution method is a clever technique used in integration to simplify complex expressions. When you encounter an intricate integral, this method can transform it into something more manageable. The key is to identify a part of the integrand that can be substituted.
In this problem, the integral \( \int \frac{x}{x^2+1} \, dx \) is solved by substituting:

• Let \( u = x^2 + 1 \), which transforms the denominator into a single variable.
• Then, differentiate \( u \) to get \( du = 2x \, dx \), or rewritten, \( x \, dx = \frac{1}{2} du \).

Substituting these, the integral simplifies drastically:
\[ V = \pi \int \frac{1}{u} \, du \]
The substitution aids in turning a seemingly complex integral into a basic logarithmic form \( \ln|u| \), making it straightforward to solve and evaluate between the given bounds.