Problem 11 - Essential Calculus

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\). $$ y=x, \quad y=3, \quad x=0 $$

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the region bounded by the line \(y=4\), and the graphs of the equations \(y=x\), \(y=3\), and \(x=0\), is \(21\pi\) cubic units.

Step 1: Identify the bounded region

The bounded region enclosed by the graphs is a triangle in the first quadrant, bounded by \(y = x\), \(y = 3\), and \(x = 0\). When this bounded region is revolved about the line \(y = 4\), it forms a solid.

Step 2: Setup the integral for the Disk Method

The disk's radius will vary as we move along the line of revolution, its radius being the difference between the line \(y = 4\) and the upper boundary of the region \(y = x\), thus the radius is given by \(4 - x\). Since we are integrating parallel to the x-axis, the differential element is \(dx\). The volume of the solid of revolution is then calculated as: \[ V = \pi \int_{a}^{b} [f(x)]^2 dx \] Where \(f(x) = 4 - x\), \(a=0\), and \(b=3\). This is because the solid rotation stops at \(x = 3\) where \(y = x\) meets \(y = 3\).

Step 3: Evaluate the integral

Substitute the limits and the function into the volume integral formula: \[ V = \pi \int_{0}^{3} (4-x)^2 dx \]. To evaluate the integral, first expand the squared term: \[ V = \pi \int_{0}^{3} (16 - 8x + x^2) dx \]. Now integrate term by term: \[ V = \pi [16x - 4x^2 + \frac{1}{3}x^3]_{0}^{3} \]. Applying the Fundamental Theorem of Calculus gives: \[ V = \pi [ (16*3 - 4*3^2 + \frac{1}{3}*3^3) - (16*0 - 4*0^2 + \frac{1}{3}*0^3)] \]

Step 4: Calculate the Volume

Simplify the expression to find the volume of the solid of revolution: \[ V = \pi [ (48 - 36 + 9) - 0] = \pi [21] = 21\pi \] cubic units.

Key Concepts

Disk Method

Imagine a solid object formed by rotating a flat shape around an axis. To find the volume of this object, the disk method is an elegant technique that slices the solid into a series of thin, circular disks (or washers if there’s a hole in the middle). These disks, when stacked along the axis of rotation, recreate the original solid.

The key to using the disk method is to identify the radius of each disk which varies as you move along the axis of rotation. The volume of each tiny disk is expressed as \(\pi r^2dx\), where \(r\) is the radius and \(dx\) is a small thickness of the disk. To find the total volume, we integrate this volume element over the entire range of the solid, summing up the volume of all disks.

Calculus Integration

Integration in calculus is the reverse operation of differentiation – it's a way to accumulate the tiny changes represented by a derivative, to get back to the quantity we started with.

In essence, while derivatives provide us with the rate at which a function changes, integrals tally up all those tiny changes to give us total change over an interval. There are many applications for integrals beyond just finding areas under curves; they are used for computing volumes, solving certain equations, and in physics, for evaluating the total work done by a force.

Understanding integrals involves recognizing the definite and indefinite forms, which are both crucial in the realm of calculus. The indefinite integral, or antiderivative, is a family of functions representing all the possible primitives of the function being integrated, while the definite integral calculates the net area under the curve within certain limits.

Solid of Revolution

A solid of revolution is a three-dimensional object obtained by revolving a two-dimensional shape around an axis. The resulting solid has a symmetry about this axis, resembling shapes like cylinders, cones, and spheres when basic functions are revolved.

These solid figures are ubiquitous in the real world – think of a wine glass formed by rotating its side profile, or an industrial tank made by spinning the cross-section of the tank around its central axis. Calculus comes into play by providing us the tools to calculate properties of these solids, such as volume, without having to physically slice and measure each piece.

Definite Integral

The definite integral is a core concept of calculus that represents the accumulation of quantities. When we speak of the definite integral of a function between two points, we're essentially looking at the total sum of values taken by that function, considering every point within the given interval.

When calculating areas or volumes, the definite integral tells us the 'net' area or volume between the curve and the axis over a specific section. The Fundamental Theorem of Calculus provides the groundwork for computing definite integrals, by tying together antiderivatives and the limits of the integral. Mathematically, the definite integral is expressed as \(\int_{a}^{b} f(x) dx\), where \(a\) and \(b\) are the boundaries of integration and \(f(x)\) is the function being integrated.