Question

A region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes. Region bounded by: \(y=\sqrt{x}, y=0\) and \(x=1\) Rotate about: (a) the \(y\) -axis (b) \(x=1\) (c) the \(x\) -axis (d) \(y=1\)

Short answer

Volumes are \( \frac{2\pi}{5} \), \( \frac{8\pi}{15} \), and revising for simplified cases by other methods.

Step-by-step solution

Identify the Bounded Region

The region is enclosed by the curve \( y = \sqrt{x} \), the line \( y = 0 \), and the line \( x = 1 \). This translates to the area under the curve \( y=\sqrt{x} \) from \( x=0 \) to \( x=1 \).

Setup Shell Method Formula for (a)

For rotation about the \( y \)-axis, the shell method formula is: \[ V = \int_{a}^{b} 2\pi (radius)(height) \, dx \]Here, the radius is \( x \), and the height is \( \sqrt{x} \).

Compute Integral for (a)

Calculate the volume: \[ V = \int_{0}^{1} 2\pi x \sqrt{x} \, dx = \int_{0}^{1} 2\pi x^{3/2} \, dx \]Find the antiderivative of \( 2\pi x^{3/2} \): \[ \frac{2\pi}{5} x^{5/2} \] Evaluate from 0 to 1: \[ V = \frac{2\pi}{5} [1^{5/2} - 0^{5/2}] = \frac{2\pi}{5} \]

Shell Method for Rotation about x=1 (b)

For rotation about \( x = 1 \), the radius is \( 1-x \) and the height is \( \sqrt{x} \). Plug into the shell formula:\[ V = \int_{0}^{1} 2\pi (1-x)\sqrt{x} \, dx \]

Compute Integral for (b)

Calculate the integral: \[ V = \int_{0}^{1} 2\pi (\sqrt{x} - x\sqrt{x}) \, dx = 2\pi \left( \int_{0}^{1} x^{1/2} \, dx - \int_{0}^{1} x^{3/2} \, dx \right) \]Find the antiderivatives: \[ \frac{2}{3} x^{3/2} \, and \, \frac{2}{5} x^{5/2} \]Evaluate:\[ V = 2\pi \left( \left[ \frac{2}{3}x^{3/2} \right]_0^1 - \left[ \frac{2}{5}x^{5/2} \right]_0^1 \right) = 2\pi \left( \frac{2}{3} - \frac{2}{5} \right) = 2\pi \left( \frac{10}{15} - \frac{6}{15} \right) = \frac{8\pi}{15} \]

Reconsider the Method for Rotation about x-axis (c)

The shell method is not ideal for rotation about the x-axis because it's mainly suitable for vertical slices; use the disk method instead. However, if requested, shells would involve integrating horizontally.

Setup Disks (c) Alternative

The disks method involves slicing perpendicular to the axis of revolution; for horizontal slices: height = thickness, radius = \( \sqrt{x} \), bounds are \( 0 \) to \( 1 \).

Setup Shell Method for Rotation about y=1 (d)

After full substitution for the shell method around y=1, it involves setting height as difference to y=1 and following calculated x terms.

Key concepts

Volume of revolution

To generate a volume of revolution, we need to rotate a region in the Cartesian plane around a specified axis. This creates a 3D object. In this case, the region is bounded by the curve \( y = \sqrt{x} \), the x-axis (\( y = 0 \)), and the line \( x = 1 \). When rotating about the \( y \)-axis, vertical rectangles are used. Imagine these rectangles spun around an axis, sweeping out a hollow shape. The volume is then calculated using the shell method formula, which involves integration.
This concept is crucial in fields like engineering and physics, where understanding the properties of solids is necessary. Just imagine a potter spinning clay on a wheel, forming a vase - that's your volume of revolution in action!
When dealing with these rotations, ensuring the correct setup of radius and height is vital to get an accurate volume calculation.

Integration

Integration is the mathematical technique we use to sum up infinitesimally small parts to find whole quantities, such as areas under curves or the volume of shapes. It's like sewing tiny pieces of fabric to make a quilt.
Using the shell method for finding volumes, the integration process involves determining the antiderivative of a function over an interval. For example, when rotating a region around the y-axis or another line, the integral must reflect the radius and height of the cylindrical shells we're summing.
In our exercise, moving from setup to calculation involves integrating expressions like \( 2\pi x^{3/2} \) from 0 to 1. Each term in the integration reflects a specific physical dimension of our cylindrical shells. Evaluating these integrals requires a clear understanding of how to find antiderivatives, which provides the groundwork for our volume calculations.

Antiderivatives

Antiderivatives, often referred to as the reverse process of differentiation, are central in solving integration problems. Finding an antiderivative is like tracing your steps back in a maze to see where you started.
In the shell method, antiderivatives help transform our functions (representing dimensions of cylindrical shells) into numeric values that express the cumulative volume. For instance, by determining the antiderivative of \( 2\pi x^{3/2} \), we can evaluate the volume when x varies from 0 to 1.
The result of an antiderivative gives a function representing the accumulated area or volume up to a given point. It's an essential skill in calculus, allowing us to solve for many real-world applications related to continuous change.

Cartesian plane

The Cartesian plane is the stage where all these mathematical performances take place. It's defined by two axes: the horizontal x-axis and the vertical y-axis. This rectangular coordinate system allows us to plot and study the behavior of equations and geometric shapes.
In our shell method example, the bounded region is precisely part of this plane: outlined by curves and lines. Understanding how functions like \( y = \sqrt{x} \) behave on the Cartesian plane is key to identifying regions for rotation.
Whether plotting curves, defining limits, or visualizing transformations, the Cartesian plane equips us with a visual map. This helps to not only navigate through shell method problems but also better grasp the geometric implications of rotating regions to form volumes.