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Chapter 14: Problem 28

Use a computer algebra system first to plot and then to approximate (with four-place accuracy) the area of the part of the given surface \(S\) that lies above the square in the \(x y\) -plane defined by: (a) \(-1 \leqq x \leqq 1,-1 \leqq y \leqq 1\) : (b) \(|x|+|y| \leqq 1\) \(S\) is the sphere \(x^{2}+y^{2}+z^{2}=4\).

Short Answer

Expert verified
Approximate numerically for regions (a) and (b) using a computer algebra system.

Step by step solution

01

Understand the Problem

We need to calculate the area of the spherical surface that lies above two different regions in the xy-plane. Our surface is defined by the sphere equation \(x^2 + y^2 + z^2 = 4\). We are given two regions: (a) \(-1 \leq x \leq 1, -1 \leq y \leq 1\), and (b) \(|x| + |y| \leq 1\).
02

Express Surface in Terms of z

The sphere's equation can be rearranged to express \(z\) in terms of \(x\) and \(y\): \(z = \sqrt{4 - x^2 - y^2}\). This is only valid for the upper hemisphere since we're looking for the area above the xy-plane.
03

Set Up the Integral Expressions

The differential area element on the surface is given by \(dA = \sqrt{1 + (z_x)^2 + (z_y)^2} \, dx \, dy\). Compute partial derivatives: \(z_x = \frac{-x}{\sqrt{4-x^2-y^2}}\) and \(z_y = \frac{-y}{\sqrt{4-x^2-y^2}}\). Substitute these into the area element expression.
04

Calculate the Area for Region (a)

Region (a) is the square defined by \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\). The area integral becomes \(\iint \sqrt{1 + \left(\frac{x^2}{4-x^2-y^2}\right) + \left(\frac{y^2}{4-x^2-y^2}\right)} \, dx \, dy\). Evaluate this integral numerically using a computer algebra system.
05

Calculate the Area for Region (b)

Region (b) is defined by \(|x| + |y| \leq 1\), forming a diamond shape in the xy-plane. Set the limits of integration accordingly and solve the integral \(\iint_{|x| + |y| \leq 1} \sqrt{1 + \left(\frac{x^2}{4-x^2-y^2}\right) + \left(\frac{y^2}{4-x^2-y^2}\right)} \, dx \, dy\) using numerical methods.
06

Approximate Values

Use numerical integration techniques available in computer algebra systems to approximate the values of the integrals from Steps 4 and 5 for four-place accuracy.

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Key Concepts

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

Surface Area
In multivariate calculus, the concept of surface area expands beyond flat surfaces to include more complex shapes like spheres. Calculating the surface area involves integrating an area element over the surface. In our problem, the surface in question is defined as part of a sphere. The sphere given is described by the equation: \[ x^2 + y^2 + z^2 = 4 \]To find the surface area that lies above a certain region in the xy-plane, we first have to express the surface in terms of either \(x\) and \(y\). Then, we compute the differential area element \(dA\), which involves partial derivatives of the surface equation. Finally, by integrating \(dA\) over the specified region, we are able to compute the desired surface area.
Spherical Coordinates
Spherical coordinates offer a convenient way of handling points on spheres or spherical surfaces. While Cartesian coordinates (\(x, y, z\)) are useful for many purposes, spherical coordinates (\(r, \theta, \phi\)) correspond naturally to spheres.In spherical coordinates, any point on a sphere can be represented as:
  • \(r\): the radius of the sphere (constant for a given sphere)
  • \(\theta\): the azimuthal angle in the xy-plane from the positive x-axis
  • \(\phi\): the polar angle from the positive z-axis
For the given sphere \(x^2 + y^2 + z^2 = 4\), we found it more straightforward to use Cartesian for initial calculations, but understanding spherical coordinates is crucial in many applications.
Numerical Integration
Numerical integration becomes vital when analytical solutions are too complex or impossible to derive. For calculating areas over regions that don't have simple bounds or do not result in easy integrals, numerical methods pave the way. In our exercise, a computer algebra system (CAS) is utilized to compute the intricacies of the double integral necessary to find the area above specific regions on the sphere. Numerical integration techniques such as trapezoidal or Simpson's rules might be employed by the CAS to numerically approximate the surface area with high precision, ensuring results with four-place accuracy.
Computer Algebra Systems
Computer Algebra Systems (CAS) are powerful tools in multivariate calculus. Their ability to carry out symbolic computations as well as numerical ones makes them invaluable, especially for more complex tasks. In problems involving calculations of surface areas over certain bounds, a CAS can:
  • Symbolically simplify expressions
  • Numerically integrate complex functions with precision
  • Visualize surfaces and areas for better comprehension
Using a CAS eliminates potential human error in manual calculations and provides quick approximations, allowing students and professionals alike to focus more on understanding geometry and exploring different mathematical scenarios.

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Most popular questions from this chapter

A uniform rectangular plate with base length \(a\). height \(b\). and mass \(m\) is centered at the origin. Show that its polar moment of inertia is \(l_{0}=\frac{1}{2} m\left(a^{2}+b^{2}\right)\).

Show by direct computation that the centroid of the triangle with vertices \((0,0),(r, 0)\), and \((0, h)\) is the point \((r / 3, h / 3)\). Verify that this point lies on the line from the vertex \((0,0)\) to the midpoint of the opposite side of the triangle and two-thirds of the way from the vertex to the midpoint.

Find the area that is cut from the surface \(z=x^{2}-y^{2}\) by the cylinder \(x^{2}+y^{2}=4\)

Solve for \(x\) and \(y\) in terms of \(u\) and \(v\). Then compute the Jacobian \(\partial(x, y) / \partial(u, v)\) \(u=\frac{2 x}{x^{2}+y^{2}}, \quad v=-\frac{2 y}{x^{2}+y^{2}}\)

Sketch the solid bounded by the graphs of the given equations. Then find its volume by triple integration. $$y=z^{2}, z=y^{2}, x+y+z=2, x=0$$
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