Question

The top outer edge of a solid with vertical sides and resting on the \(x y\) -plane is modeled by \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+\left(1+\sin ^{2} 2 t\right) \mathbf{k},\) where all measure- ments are in centimeters. The intersection of the plane \(y=b(-3

Short answer

To answer the exercise, a computer algebra system is used to graph the solid and to calculate the surface area and the volume. The graph visualizes the shape of the solid. The surface area is calculated by integrating the magnitude of the cross-product of the derivative of the function with the vector pointing up. Lastly, the volume is determined by performing a triple integration using cylindrical coordinates. These values are then approximated in the computer algebra system.

Step-by-step solution

Graph the solid

To graph, one can use the parametric equation \(\mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + (1 + \sin^{2} 2 t) \mathbf{k}\) to generate the 3D plot in a computer algebra system. In the system, input the equation and specify that \(t\) varies from -\(\pi\) to \(\pi\). This will produce a 3D object on the plane.

Approximate the lateral surface area

To approximate the lateral surface area, one would need to integrate the magnitude of the cross-product of the derivative of the function with the vector pointing up. The surface area \(S\) of a parametrically defined surface given as \(\mathbf{r}(u,v)\) is calculated as \(S = \int \int |\mathbf{r}_{u} \times \mathbf{r}_{v}| du dv\). In this case, \(u = t\), \(v = b\), and \(\mathbf{r} = (3 \cos t, 3 \sin t, 1+ \sin^2 2t)\). After calculating the necessary derivatives and the cross product, one can perform the double integral operation in the computer algebra system to obtain the estimated lateral surface area

Calculate the volume

To calculate the volume, one will need to use triple integration. Because the solid rests on the \(xy\)-plane, one can take advantage of the cylindrical coordinates system. The volume \(V\) can be found as \(V = \int \int \int dV\), where \(dV = r d\theta dz dr\). Here, \(r = \sqrt{3^2\cos^2 t + 3^2 \sin^2 t} = 3\), \(\theta = t\), and \(z\) ranges from 0 to \(1 + \sin^2 2t\). Substituting in the cylindrical coordinates and evaluating the triple integral in the computer algebra system will give the volume of the solid.

Key concepts

3D Parametric Plot

Creating a 3D parametric plot is an essential skill in visualizing complex geometric shapes in calculus. It involves setting a three-dimensional space where each point is determined by a set of three functions, each one depending on one or more parameters. In the exercise task, the solid's edge is defined by a parametric equation involving trigonometric functions.

The equation given, \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+\left(1+\sin ^{2} 2 t\right) \mathbf{k}\), encapsulates how each coordinate changes with parameter \(t\). By manipulating \(t\) within the specified range of \(t\) from -\(\pi\) to \(\pi\), the computer system draws the solid's edge in 3D space. This can be done using software like MATLAB, Maple, or Mathematica, graphing a curve that wraps around creating the visual representation of the solid. Such visual aids are vital for students to understand the shape and structure of the object they're studying before moving on to more complex calculations, such as determining surface area or volume.

The 3D plot is directly tied to the parametric equations, providing an invaluable tool for predicting the behavior of the modeled solid, analyzing its symmetry, and understanding its geometric properties.

Lateral Surface Area Integration

In calculus, finding the lateral surface area of a solid involves integration over a surface. The exercise problem asks for an approximation of the lateral surface area using a parametric representation of the solid's boundary. When the surface is defined by a vector function \(\mathbf{r}(u,v)\), the area can be calculated by integrating the magnitude of the normal vector over the region of interest.

The procedure requires us to first find the partial derivatives of \(\mathbf{r}(u,v)\) with respect to each parameter. These derivatives \(\mathbf{r}_u\) and \(\mathbf{r}_v\) are then used to compute the cross product \(\mathbf{r}_u \times \mathbf{r}_v\), which gives us a vector that is normal to the surface at each point. The magnitude of this normal vector, \( |\mathbf{r}_u \times \mathbf{r}_v| \), provides the differential area element at each point. Integrating this over the entire surface gives us the total lateral surface area.

In the provided exercise's context, because we only have a function of \(t\), the cross-product simplifies to the derivative of the vector function with respect to \(t\), and a unit vector pointing upward. By integrating this magnitude from \(t = -\pi\) to \(t = \pi\) and over the range of \(b\), one can use computational tools to approximate the lateral surface area. This technique demonstrates the power of integration in calculating areas that are not easily decomposable into simple geometric shapes.

Volume Calculation Triple Integration

Calculating the volume of a solid with complex geometry can often be accomplished through triple integration. In calculus, triple integration extends the concept of single integrals to find volumes by integrating over three-dimensional regions. The volume \(V\) conceptually represents the accumulation of infinitesimally small volume elements, \(dV\), over the space occupied by the solid.

In the case of the solid resting on the \(xy\)-plane, we can convert the problem into cylindrical coordinates to simplify integration. For a function in the form of \(\mathbf{r}(t)\), we identify how the \(r\), \(\theta\), and \(z\) coordinates change in the 3D space. The volume element in cylindrical coordinates is given by \(dV = r d\theta dz dr\).

When setting up our integral boundaries, \(r\) is constant at 3 due to the symmetry of the solid with respect to the origin, \(\theta\) or \(t\) varies between -\(\pi\) and \(\pi\), and \(z\) ranges from 0 to \(1 + \sin^2 2t\) following the top edge of the solid described in the parametric equation. The triple integral, \(V = \int \int \int dV\), is then carried out over these limits. This approach is particularly efficient for shapes that lack simple geometrical definitions, allowing for precise volume calculations. With the aid of a computer algebra system, students can perform the integration to find the volume without manually solving the complex integrals.

By understanding and applying triple integration, students can tackle a wide range of real-world problems involving volume calculations, from engineering designs to natural phenomena modeling.