Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(x=6\). $$ y=6-x, \quad y=0, \quad y=4, \quad x=0 $$

Short answer

The volume of the solid generated by revolving the region about the line \(x=6\) is \(6\pi\) cubic units.

Step-by-step solution

Identify the Bounded Region

Sketch the graphs of the equations \(y=6-x\), \(y=0\), \(y=4\), and \(x=0\). The bounded region is a triangle with vertices at (0,0), (0,4), and (2,4).

Setup for Disk/Washers Method

The rectangle will be revolved about the line \(x=6\). To apply the disk/washers method, we need to identify the radius and thickness of the disk(s). In this problem, the thickness (dx) is along the x-axis. The outer radius is the difference between the line of revolution and the far edge of the region which is \(5 - y\) and the inner radius is the distance from the line of revolution to the close edge of the region which is \(6 - (6 - x)\), simplifying to \(x\).

Setup the Integral

The volume V of the solid is given by the difference in volumes generated by the outer and inner radii, integrated from 0 to 2 (the x-coordinates of the bounded region). In mathematical terms: \(V = \pi \int_{0}^{2} [(5-y)^2 - (x)^2] dx\).

Substitute for y

Rewrite the integral in terms of x. Since \(y = 6 - x\), the volume integral becomes \(V = \pi \int_{0}^{2} [(5 - (6 - x))^2 - (x)^2] dx = \pi \int_{0}^{2} [(1 + x)^2 - x^2] dx\). This simplifies to \(V = \pi \int_{0}^{2} (2x + 1) dx\).

Evaluate the Integral

Compute the integral to find the volume of the solid, \(V = \pi [x^2 + x]_{0}^{2} = \pi[4 + 2 - 0] = 6\pi\).

Key concepts

Disk/Washer Method

When calculating the volume of a solid of revolution, the disk/washer method is a handy technique that provides a straightforward approach. This method slices the solid into thin disks or washers perpendicular to the axis of revolution.

Picture slicing a carrot or a cucumber into thin round pieces — this is similar to how we're slicing our solid, but in a mathematical sense.

Disk vs. Washer

The difference between a 'disk' and a 'washer' is simply whether the slice is solid or has a 'hole' in the middle, respectively. Disks are used when the solid is full at the slice, while washers are used when there is empty space, like a doughnut.

• Disk: A solid circular slice with no central hole.
• Washer: A circular slice with a central hole, where the outer and inner radii come into play.

In the given problem, we visualize disks (or washers) around the line x=6, so each slice's volume is determined by the radii and thickness of the disks/washers. The volume calculation will be the sum of all the disk/washer volumes, which is efficiently done using integral calculus.

Integral Calculus

Concrete results from abstract principles — that's what integral calculus can help us achieve. In the context of finding volumes, integrals accumulate quantities, which is how we can sum up the infinitely many disk/washer slices to find the whole volume.

Think of it as a magical machine: You feed it a function (the shape of a disk or washer), and it tells you the accumulated sum, such as total distance from a speed function, or in our case, the total volume from area slices.

Applying the Integral

The given function, area, or radius is integrated over the interval that represents the bounds of our region of interest. This bounded region is from where one slice starts (in our exercise, it's at x=0) to where it ends (the triangle's far edge is at x=2).

The integral is, quite literally, the continuous sum of all these infinitesimally small slices between the boundaries, multiplied by the thickness of each slice to give us a volume. It allows the precise calculation of the volume of oddly shaped solids, which would be difficult, if not impossible, by simple geometric formulas.

Volume Calculation

Put simply, volume is the measure of how much space a three-dimensional object occupies. The goal is to calculate the space within the bounds of the solid of revolution.

Calculating Volume Using the Integral

This is done by integrating the area of each slice — or when we're looking at a graph, integrating a function that represents the radius squared (since the area of a circle is \( \pi r^2 \)) times pi over the given interval. This method aligns with the principle that the whole is equal to the sum of its parts.

To go from the simple equation of a circle's area to calculating the complex volume of a solid generated by revolving a region around an axis, we rely on integral calculus. The careful setup and computation of the integral that represents all these slices stacked next to each other give us the total volume.

Always remember to substitute the variable correctly if the boundary equations are not in terms of the integration variable. In our exercise, we integrated with respect to x and replaced y with \(6-x\) as needed, to ensure that the integral could be evaluated correctly.