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Chapter 15: Problem 9

Use a computer algebra system to evaluate \(S: z=10-x^{2}-y^{2}, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 2\)

Short Answer

Expert verified
The volume of the solid represented by the function \(z = 10 - x^{2} - y^{2}\) for \(0 \leq x \leq 2\) and \(0 \leq y \leq 2\) is a cap of a sphere located at (0,0,10) with a radius of sqrt(10) and cut off by the x-y plane. A Computer Algebra System would be used to calculate and visualize it.

Step by step solution

01

Identify the function

This problem is asking us to evaluate the function \(S: z=10-x^{2}-y^{2}\), for \(0 \leq x \leq 2\) and \(0 \leq y \leq 2\). This is actually an expression for the volume of a solid in three dimensions.
02

Apply the limits

Now, we need to apply these limits to x and y in the expression. This means we have to insert each corresponding value of x and y, starting from the lower bound (0) going up to the upper bound (2) into the formula of z and calculate the consequent value of each.
03

Solve for z

Take every pair (x,y) from the drawn square (since both x and y move between 0 and 2) and put it into the equation of z. For this, a computer algebra system could be used to create a matrix that contains the values of z corresponding to the values of x and y.
04

Interpret results

Finally, interpret these results. Every value of z that the algebra system gives is a point on the surface S in the three-dimensional (x,y,z) space. Notice, all those points form a cap of a sphere located at (0,0,10) with a radius of sqrt(10) and cut off by the x-y plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Computer Algebra System
A Computer Algebra System (CAS) is a powerful tool that helps in solving mathematical problems. It's like having a calculator that can handle algebraic expressions, perform calculus operations, and more. However, a CAS is more advanced and can perform symbolic manipulation, which means it can solve equations, factor expressions, and evaluate complex functions with precision.

In the context of evaluating the function \(S: z=10-x^{2}-y^{2}\), a CAS can quickly compute the expression for every pair of values within a specific range. You input the formula, along with the constraints \(0 \leq x \leq 2\) and \(0 \leq y \leq 2\), and the CAS will output values for \(z\). Instead of manually calculating each point, the CAS provides an efficient and error-free process, transforming data interpretation into a more manageable task.

Benefits of using a CAS include:
  • Speed: Rapid calculations without errors.
  • Accuracy: Precision in handling complex algebraic computations.
  • Visualization: Often comes with graphing tools to illustrate functions.
Three-dimensional Graphing
Three-dimensional graphing is a method to visualize mathematical functions and surfaces in a space that has width, depth, and height. For many students, seeing graphs in 3D can make it easier to understand relationships between variables. With this approach, you aren't limited to just seeing a line or a curve; you can visualize solids or surfaces that occupy space.

To graph the function \(S: z=10-x^{2}-y^{2}\) using three-dimensional graphing, you'd set up axes for \(x\), \(y\), and \(z\). Typically, software or graphing tools can plot this by calculating \(z\) for each combination of \(x\) and \(y\) over the defined range. This creates a surface display, helping to show how the values change together.

Three-dimensional graphing provides insights such as:
  • Shape: Understanding curves or caps like the one formed by our sphere segment.
  • Volume: Visualizing enclosed areas or limits of functions.
  • Intersections: Finding where surfaces meet or overlap.
Surface in Three Dimensions
A surface in three dimensions is defined by an equation involving three variables, typically \(x\), \(y\), and \(z\). These surfaces can take on various shapes such as planes, spheres, and more complex forms. In the case of our function \(z=10-x^{2}-y^{2}\), the surface is a part of a sphere, sometimes called a spherical cap.

This surface is visualized by how \(z\) changes as \(x\) and \(y\) vary within their confined intervals鈥攆rom 0 to 2 in our example. The equation leads to a three-dimensional form, illustrating how \(z\) decreases as \(x\) and \(y\) increase. Notably, it portrays a cap, since the original equation \(z=10-x^{2}-y^{2}\) relates to the top portion of a sphere of radius \(\sqrt{10}\), centered at (0,0,10) and extending back to the \(xy\)-plane.
  • The equation defines the height \(z\) over the \(xy\)-plane within the bounded square.
  • The graph shows curvature, typical of a spherical body section.
  • This function's shape and its limits help verify volume or surface calculations.

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Most popular questions from this chapter

$$ \text { Define a parametric surface. } $$

Let \(\quad \mathbf{E}=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k} \quad\) be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\) and its circular base in the \(x y\) -plane.

Let \(\mathbf{G}(x, y)=\left(\frac{-y}{x^{2}+4 y^{2}}, \frac{x}{x^{2}+4 y^{2}}, 0\right)\). Prove or disprove that there is a vector-valued function \(\mathbf{F}(x, y, z)=(M(x, y, z), N(x, y, z), P(x, y, z))\) with the following properties. (i) \(M, N, P\) have continuous partial derivatives for all \((x, y, z) \neq(0,0,0) ;\) (ii) \(\operatorname{Curl} \mathbf{F}=\mathbf{0}\) for all \((x, y, z) \neq(0,0,0) ;\) (iii) \(\mathbf{F}(x, y, 0)=\mathbf{G}(x, y)\).

Find the area of the surface over the given region. Use a computer algebra system to verify your results. $$ \begin{aligned} &\text { The part of the paraboloid } \mathbf{r}(u, v)=4 u \cos v \mathbf{i}+4 u \sin v \mathbf{j}+\\\ &u^{2} \mathbf{k} \text { , where } 0 \leq u \leq 2 \text { and } 0 \leq v \leq 2 \pi \end{aligned} $$

Use the Divergence Theorem to evaluate \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S\) and find the outward flux of \(\mathrm{F}\) through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. $$ \begin{aligned} &\mathbf{F}(x, y, z)=x y z \mathbf{j}\\\ &S: x^{2}+y^{2}=9, z=0, z=4 \end{aligned} $$
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