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: \(y=4-x^{2}\) and \(y=0\) Rotate about: (a) \(x=2\) (b) \(x=-2\) (c) the \(x\) -axis (d) \(y=4\)

Short answer

Use the Shell Method around x=2 and x=-2; Disk for x-axis; Shell for y=4.

Step-by-step solution

Understand the Region

The region we're working with is bounded by the parabola \( y = 4 - x^2 \) and the line \( y = 0 \). This essentially means we are considering the top half of the parabola, where it opens downward from its vertex at \((0, 4)\) and intersects the x-axis at \((-2, 0)\) and \((2, 0)\).

Identify the Shell Method Formula

The Shell Method formula for finding the volume of a solid of revolution revolving around a vertical line is:\[V = 2\pi \int_a^b (radius)(height)\, dx\]

Solve (a) for Rotation about \(x = 2\)

The radius from a point \((x, y)\) to the line \(x = 2\) is \(|x - 2|\). The height of the shell is given by the function \(y = 4 - x^2\). Therefore, the volume is:\[V = 2\pi \int_{-2}^{2} (2 - x)(4 - x^2)\, dx\]Calculate the integral to find:\[V = 2\pi \left[\int_{-2}^{2} (8 - 4x^2 - 2x + x^3)\, dx\right]\] Then evaluate the definite integral.

Solve (b) for Rotation about \(x = -2\)

The radius from a point \((x, y)\) to the line \(x = -2\) is \(|x + 2|\). Thus, the volume becomes:\[V = 2\pi \int_{-2}^{2} (x + 2)(4 - x^2)\, dx\]Solve it similarly as in (a) by evaluating the integral:\[V = 2\pi \left[\int_{-2}^{2} (4x + 8 - x^3 - 2x^2)\, dx\right]\].

Solve (c) for Rotation about the x-axis

Here we can't use the Shell Method directly since that is typically for revolving around a vertical line. Instead, use the Disk/Washer Method. The formula in this context changes as shape depends height-wise:\[V = \pi \int_{a}^{b} ((outer\ radius)^2 - (inner\ radius)^2)\, dy\]Convert \(x\) terms to \(y\) terms:\( x = \sqrt{4 - y} \).The volume can be found using:\[V = 2\pi \int_{0}^{4} (\sqrt{4-y}) \cdot y\, dy\].

Solve (d) for Rotation about \(y = 4\)

The radius from a point \((x, y)\) to the line \(y = 4\) is \(|4 - (4 - x^2)| = x^2\). So, the volume here is:\[V = 2\pi \int_{-2}^{2} x^2\cdot x^2 \cdot dx\]Simplify to:\[V = 2\pi \int_{-2}^{2} x^4\, dx\], and find the definite integral for the volume.

Key concepts

Volume of Solids of Revolution

The volume of solids of revolution is a technique in integral calculus used to compute volumes of three-dimensional objects created by rotating a two-dimensional shape around an axis. This method is particularly useful when the object has symmetry due to rotation. It involves imagining a shape, like a region on the Cartesian plane, revolving around a line (the axis of revolution) to create the solid.
In the context of our exercise, we're dealing with a parabola and a line forming a bounded region. When this region is revolved around different axes, it transforms into a solid, for which we need to determine the volume. By using methods like the Disk/Washer Method or the Shell Method, we multiply the infinitesimal area of cross sections or shells by a rotational distance to compute the volume efficiently. The Shell Method, in particular, is handy when revolving around vertical lines.
Remember, the choice between using shells, disks, or washers often depends on the axis of rotation and the orientation of the region being rotated.

Integral Calculus

Integral calculus is a fundamental part of calculus that deals with the concept of integration. Integration allows us to find quantities like areas under curves, lengths of curves, or, in our case, volumes of solids. It's the reverse process of differentiation, focusing on accumulation of quantities.
In solving our problem with shells, we're essentially summing up an infinite number of infinitesimally thin cylindrical shells to find the total volume. This is captured through the integral:

• Set the shell radius and height based on the described functions and axis of rotation.
• Establish the integration limits, which in this exercise, are from the points where the function intersects the x-axis.
• Plug these into the integral formula to compute the desired quantity.

Integral calculus serves as a powerful tool in transcending direct measurement, using mathematical functions to calculate large-scale quantities like volume.

Cartesians Plane

The Cartesian plane is a two-dimensional coordinate system defined by an x-axis and a y-axis. It allows us to describe locations and regions mathematically using coordinates (x, y). It's instrumental in solving a multitude of problems in mathematics, including the volume of solids of revolution.
In the problem we're addressing, the Cartesian plane helps us define the bounded region of interest. The parabola represented by the equation \( y = 4 - x^2\) and the line \( y = 0\) depict the top half of the parabola stretching from \((-2, 0)\) to \((2, 0)\). These intersections show us where the region touches the x-axis, providing essential boundaries for integration limits.
The Cartesian plane's coordinate system gives us the necessary framework to apply calculus concepts and perform rotations around axes efficiently. It turns abstract curves and lines into tangible coordinates, aiding in visualization and calculation.

Parabola

A parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. In a simple quadratic equation \( y = ax^2 + bx + c \), the parabola can open upwards or downwards, depending on the sign of the leading coefficient \( a \).
In the context of our exercise, the parabola \( y = 4 - x^2 \) opens downwards and is symmetric around the y-axis. The vertex of this parabola, the highest point due to its downward opening nature, is at \((0, 4)\). It intersects the x-axis at \( x = -2 \) and \( x = 2 \), which are the roots of the equation, indicating where the parabola crosses the x-axis.

• The parabola's shape is crucial when determining the height of the cylindrical shells during integration.
• It provides the necessary function that defines the upper limit of integration in the axis rotations.

Understanding parabolas and their properties enable us to apply precise mathematical techniques to solve real-world volume calculation problems efficiently.