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Find the surface area of the part of the sphere x² + y² + ² = 100 that lies above the cone z =

Find the surface area of the part of the sphere x² + y² + ² = 100 that lies above the cone z =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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how do i solve this calculus problem?

**Problem Statement:** Find the surface area of the part of the sphere \( x^2 + y^2 + z^2 = 100 \) that lies above the cone \( z = \sqrt{x^2 + y^2} \). **Solution Explanation:** To solve this problem, follow these steps: 1. **Understand the Surfaces:** - The sphere has the equation \( x^2 + y^2 + z^2 = 100 \), representing a sphere with a radius of 10 units. - The cone has the equation \( z = \sqrt{x^2 + y^2} \), which opens upwards and is symmetric around the z-axis. 2. **Identify the Region of Interest:** - We are interested in the part of the sphere that is above the cone, meaning where the surface of the sphere is greater than the cone. This involves solving inequalities involving these equations. 3. **Set Up the Integral:** - Consider switching to spherical coordinates for integration. The equations translate into these coordinates by letting: - \( x = \rho \sin \phi \cos \theta \) - \( y = \rho \sin \phi \sin \theta \) - \( z = \rho \cos \phi \) - Here, \(\rho\) represents the radius, \(\phi\) the polar angle, and \(\theta\) the azimuthal angle. 4. **Calculate the Limits:** - Determine the limits of integration for \(\phi\) and \(\theta\) such that the inequalities are satisfied, mapping out the zone of the sphere above the cone. 5. **Evaluate the Surface Integral:** - Calculate the surface area integral over the specified region. Integrate using the spherical coordinate form of the area element on the sphere's surface. 6. **Compute the Final Answer:** - Solve the integral to find the surface area of the required segment of the sphere. This approach involves understanding and visualizing geometrical regions, transitioning to appropriate coordinate systems, and applying calculus for solving integrals on surfaces. This problem blends algebraic understanding of 3D shapes with the application of calculus, emphasizing the cross-disciplinary skills necessary for problem-solving in mathematical physics and engineering contexts.
Transcribed Image Text:**Problem Statement:** Find the surface area of the part of the sphere \( x^2 + y^2 + z^2 = 100 \) that lies above the cone \( z = \sqrt{x^2 + y^2} \). **Solution Explanation:** To solve this problem, follow these steps: 1. **Understand the Surfaces:** - The sphere has the equation \( x^2 + y^2 + z^2 = 100 \), representing a sphere with a radius of 10 units. - The cone has the equation \( z = \sqrt{x^2 + y^2} \), which opens upwards and is symmetric around the z-axis. 2. **Identify the Region of Interest:** - We are interested in the part of the sphere that is above the cone, meaning where the surface of the sphere is greater than the cone. This involves solving inequalities involving these equations. 3. **Set Up the Integral:** - Consider switching to spherical coordinates for integration. The equations translate into these coordinates by letting: - \( x = \rho \sin \phi \cos \theta \) - \( y = \rho \sin \phi \sin \theta \) - \( z = \rho \cos \phi \) - Here, \(\rho\) represents the radius, \(\phi\) the polar angle, and \(\theta\) the azimuthal angle. 4. **Calculate the Limits:** - Determine the limits of integration for \(\phi\) and \(\theta\) such that the inequalities are satisfied, mapping out the zone of the sphere above the cone. 5. **Evaluate the Surface Integral:** - Calculate the surface area integral over the specified region. Integrate using the spherical coordinate form of the area element on the sphere's surface. 6. **Compute the Final Answer:** - Solve the integral to find the surface area of the required segment of the sphere. This approach involves understanding and visualizing geometrical regions, transitioning to appropriate coordinate systems, and applying calculus for solving integrals on surfaces. This problem blends algebraic understanding of 3D shapes with the application of calculus, emphasizing the cross-disciplinary skills necessary for problem-solving in mathematical physics and engineering contexts.
Expert Solution
Step 1

Given:

The equation of sphere is:

                                   x2+y2+z2=100                      .............(1)

The equation of cone is:

                                        z=x2+y2                         ............(2)

To Find:

We have to find the surface area of the part of the sphere that lies above the cone.

Formula to be used:

Firstly calculate the intersection of the given curves.

Then the formula to calculate the surface area is given by:

                                     Area=∬1+zx2+zy2dA

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