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Find the volume of the solid that lies with thosphere X²+ y²7X²³² +2²=36 and outside the the xy plane 2=7√x²+y above come r

Find the volume of the solid that lies with thosphere X²+ y²7X²³² +2²=36 and outside the the xy plane 2=7√x²+y above come r

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Title: Calculating the Volume of a Solid in 3D Space --- **Problem:** Find the volume of the solid that lies within the sphere \( x^2 + y^2 + z^2 = 36 \), above the \( xy \)-plane, and outside the cone \( z = 7 \sqrt{x^2 + y^2} \). **Description:** 1. **Sphere Equation:** - The sphere's equation is given by \( x^2 + y^2 + z^2 = 36 \). This represents a sphere centered at the origin with a radius of 6. 2. **Constraints:** - The volume is limited to the region above the \( xy \)-plane. This restricts the volume to the upper half of the sphere. - The volume is also outside the cone defined by \( z = 7 \sqrt{x^2 + y^2} \). This is a cone with its vertex at the origin and opening upwards. 3. **Visualization:** - Imagine the sphere being sliced by the \( xy \)-plane, leaving only the upper hemisphere. - The cone, extending from the origin, creates a region that is not considered. The solid of interest is outside this conical region. **Diagrams:** - The drawing includes a sphere and a cone, but no specific diagrams or graphs were provided for this problem. It may help to sketch the sphere and the cone on a coordinate system to visualize the intersected regions clearly. **Solution Approach:** To solve this problem: - Use spherical or cylindrical coordinates to integrate over the region that satisfies all above conditions. - The boundaries and limits of integration are defined by the sphere and cone equations. **Conclusion:** This problem involves understanding the geometric intersection of a sphere and a cone in 3D space and calculating the volume of the region that satisfies the given conditions.
Transcribed Image Text:Title: Calculating the Volume of a Solid in 3D Space --- **Problem:** Find the volume of the solid that lies within the sphere \( x^2 + y^2 + z^2 = 36 \), above the \( xy \)-plane, and outside the cone \( z = 7 \sqrt{x^2 + y^2} \). **Description:** 1. **Sphere Equation:** - The sphere's equation is given by \( x^2 + y^2 + z^2 = 36 \). This represents a sphere centered at the origin with a radius of 6. 2. **Constraints:** - The volume is limited to the region above the \( xy \)-plane. This restricts the volume to the upper half of the sphere. - The volume is also outside the cone defined by \( z = 7 \sqrt{x^2 + y^2} \). This is a cone with its vertex at the origin and opening upwards. 3. **Visualization:** - Imagine the sphere being sliced by the \( xy \)-plane, leaving only the upper hemisphere. - The cone, extending from the origin, creates a region that is not considered. The solid of interest is outside this conical region. **Diagrams:** - The drawing includes a sphere and a cone, but no specific diagrams or graphs were provided for this problem. It may help to sketch the sphere and the cone on a coordinate system to visualize the intersected regions clearly. **Solution Approach:** To solve this problem: - Use spherical or cylindrical coordinates to integrate over the region that satisfies all above conditions. - The boundaries and limits of integration are defined by the sphere and cone equations. **Conclusion:** This problem involves understanding the geometric intersection of a sphere and a cone in 3D space and calculating the volume of the region that satisfies the given conditions.
Title: Calculating the Volume of a Geometrical Solid Text: "Find the volume of the solid that lies within the sphere \(x^2 + y^2 + z^2 = 36\), above the \(xy\)-plane, and outside the cone \(z = 7 \sqrt{x^2 + y^2}\)." Description: This problem involves finding the volume of a solid in three-dimensional space. The solid is defined by a sphere and a cone, with additional conditions limiting its extent to above the xy-plane. 1. **Sphere Equation**: \(x^2 + y^2 + z^2 = 36\) - Represents a sphere centered at the origin with a radius of 6. 2. **Cone Equation**: \(z = 7 \sqrt{x^2 + y^2}\) - Describes a cone with its vertex at the origin. 3. **Volume Calculation**: - The volume sought is the portion of the sphere that lies above the xy-plane and is exterior to the defined cone. To solve the problem, one would typically use integration techniques from multivariable calculus, employing spherical or cylindrical coordinates based on symmetry and the nature of the solid.
Transcribed Image Text:Title: Calculating the Volume of a Geometrical Solid Text: "Find the volume of the solid that lies within the sphere \(x^2 + y^2 + z^2 = 36\), above the \(xy\)-plane, and outside the cone \(z = 7 \sqrt{x^2 + y^2}\)." Description: This problem involves finding the volume of a solid in three-dimensional space. The solid is defined by a sphere and a cone, with additional conditions limiting its extent to above the xy-plane. 1. **Sphere Equation**: \(x^2 + y^2 + z^2 = 36\) - Represents a sphere centered at the origin with a radius of 6. 2. **Cone Equation**: \(z = 7 \sqrt{x^2 + y^2}\) - Describes a cone with its vertex at the origin. 3. **Volume Calculation**: - The volume sought is the portion of the sphere that lies above the xy-plane and is exterior to the defined cone. To solve the problem, one would typically use integration techniques from multivariable calculus, employing spherical or cylindrical coordinates based on symmetry and the nature of the solid.
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