91Ó°ÊÓ

When dealing with a volume of solids of revolution, we're often trying to find the volume of a solid that results when a given region in a plane is rotated about a specific line, known as the "axis of rotation". This means the shape is spun around a line to create a three-dimensional object. This concept is widely used in calculus to help understand and calculate the space a certain shape will occupy when in this new, 3D form.

• The region being rotated is usually bounded by one or more curves or lines.
• The axis of rotation can be either the x-axis, y-axis, or any other line parallel to these axes.

The volume of these solids can be calculated using integral calculus, where appropriate integration techniques are applied. One of the common methods used to calculate such volumes is the washer method, particularly when the solid is hollow in the middle.

The Washer Method is a technique used to find the volume of a solid of revolution, especially when the solid has a hole in the middle, like a washer. This is a refined way of using the disk method for cases where there's an inner separate boundary creating this hole.
To apply the Washer Method, follow these steps:

• Consider a horizontal or vertical slice of the solid to understand the shape of the washer. The outer and inner radii define the thickness of the solid at that slice.
• Calculate the volume of each washer by finding the outer radius and inner radius, which varies with the position along the axis of rotation.
• The formula used is: \[ dV = \pi \left((R_{ ext{outer}})^2 - (R_{ ext{inner}})^2\right) \, dy \] where the integral will sum up these volumes for either the y-axis or the x-axis rotation.

In our original problem, since we rotated around the y-axis, no inner radius was necessary because the boundary line starts directly from the axis (no hole in the middle). This simplifies to the original disk formula as there's only one defined radius \(R(y) = 3-y\).

Integral Calculus deals with the concept of integration, which is the reverse process of differentiation. It's a core concept in calculus used to find areas, volumes, and central points of more complicated shapes like curves.
Integration allows you to "add up" infinitely small parts to find a "whole". Specifically for volume problems, integration helps calculate the total space occupied by a solid of revolution.
Here’s a simple overview of how integration ties into volume problems:

• Determine the function that describes your solid or the region being rotated.
• Use definite integration to sum up all small volume elements (like disks or washers) across an interval.
• The integral \[ \int_{a}^{b} f(y) \, dy \] computes the sum for the bounds \(a\) to \(b\), which in our case corresponds to \(0\) to \(2\) for the integral of the washer method.

Evaluating this integral gives us the total volume of the solid formed by the region's rotation. In our specific problem, expanding and integrating the function \((3-y)^2\) was the core step, breaking it apart into simpler integrals to arrive at the solution.