Problem 18 - Essential Calculus

Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(x\) -axis. $$ y=x^{2}, \quad x=0, \quad y=9 $$

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the plane region about the x-axis is \(81\pi\).

Step 1: Identifying the shell's width, height and radius

In the shell method, the shells are parallel to the axis of rotation (the x-axis in this case). The width of the shells is an infinitesimally small change in x, \(dx\). The height of each shell is given by the function, \(y = x^{2}\). The radius is the x-coordinate of the function, \(x\). Hence the volume of an elemental shell is \(2\pi \cdot x \cdot x^{2} \cdot dx\).

Step 2: Setting up the integral

To find the total volume, we integrate the volume of the elemental shell with respect to \(x\) (since \(dx\) is the width of the shell) from \(x=0\) to \(x=3\) (since \(y=9\) when \(x=3\)). Hence the integral set up is \(\int_{0}^{3} 2\pi \cdot x \cdot x^{2} \cdot dx\).

Step 3: Evaluating the integral

This integral is evaluated by first simplifying to get \(\int_{0}^{3} 2\pi \cdot x^{3} \cdot dx\), which leads to \(\left[ \frac{1}{2} \cdot 2\pi \cdot x^{4} \right]_{0}^{3} = 81\pi\).

Key Concepts

Volume of Solids of Revolution

When studying calculus, particularly in the context of finding volumes, the "Volume of Solids of Revolution" is a key concept. Imagine taking a flat, two-dimensional shape and spinning it around an axis. The shape will sweep out a 3D solid. This process is called revolution.

By using methods like the disk, washer, or shell method, you can determine the volume of these solids. Each method involves slicing the solid in a systematic way:

• Disk Method: Slices are perpendicular to the axis of revolution.
• Washer Method: Similar to the disk method but accounts for a hollow center.
• Shell Method: Slices are parallel to the axis of revolution.

The shell method is used when revolving around a horizontal or vertical line and involves cylinders or shells, making it helpful when the slice pattern is not easy to describe perpendicularly. A thin cylindrical shell has a volume given by the formula:\[ V = 2\pi \cdot \text{radius} \cdot \text{height} \cdot \text{thickness} \]In our exercise, the equation \(y = x^2\) creates a curve that, when revolved around the \(x\)-axis, forms the desired solid.

Integrals in Calculus

Integrals in calculus allow us to find areas, volumes, and more, particularly those that are non-linear and could not be calculated with basic geometry. They are the reverse processes of differentiation. When you integrate a function, essentially you are summing up infinitely many bits of area, line, or volume to find a total.

In the exercise we solved, the integral comes from adding up the volumes of infinite, infinitesimally thin shells. This sum gives the entire volume. The steps for integration in this context are:

• Identify the function that describes the height of the shells, such as \(f(x) = x^2\).
• Form an integrative model by multiplying the function by the components of the shell volume formula, which includes a constant and variables defining size.
• Determine the limits of integration. They correspond to points on the curve where the region exists, bounded along the x-axis from where \(x = 0\) to where \(x = 3\).

Through the combination of these trinities of calculation, we set up the volume to be found by the integral.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus plays a crucial role in connecting the process of differentiation and integration. It forms the cornerstone that allows us to compute exact values of integrals using antiderivatives.This theorem is split into two parts:

• First Part: Asserts that the integration of a function gives its accumulation function, effectively aligning with its antiderivative or indefinite integral.
• Second Part: Allows us to evaluate the exact value of a definite integral using that antiderivative. Here, it's about evaluating the antiderivative at the boundaries of the integral.

To visualize this, consider our exercise where the polynomial \(x^2\) was integrated to become \(\frac{1}{4}x^4\). By evaluating this result from limits 0 to 3, and considering the 2\(\pi\) factor, we find the total volume. The knowledge from this theorem empowers the shell method calculations by validating their accuracy and exactness, turning abstract integration steps into tangible solutions.