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A thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p = A(r² + y²)=?, where A is a constant. (n) Show that the density expressed in spherical coordinates (r, 0, 4) is p = Ar^(1 – cos? 0) con² 0. (b) Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral. Hint: You may find the substitution u = cos 0 useful. |

A thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p = A(r² + y²)=?, where A is a constant. (n) Show that the density expressed in spherical coordinates (r, 0, 4) is p = Ar^(1 – cos? 0) con² 0. (b) Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral. Hint: You may find the substitution u = cos 0 useful. |

Elementary Geometry for College Students
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter9: Surfaces And Solids
Section9.CT: Test
Problem 8CT: Determine whether the statement is true or false. a A right circular cone has exactly two bases....
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Volume integral over a sphere in spherical coordinates The volume integral of f(r,6,ø) over a spherical region of radius R, centred on the origin, is given by I = . f6r.0.6)r² sin o dr) do) do. I = r sin 0 cos o, y =r sin 0 sin ø, z=r cos 0. 10. Spherical coordinates (r, 0, ø) are related to Cartesian coordinates (x, y, 2) by the equations 1= r sin @ cos o, y=r sin & sin o, 2=r cos 0. In spherical coordinates a volume element has volume 6V = r sin e ốr 0 độ. r sin e (a) (b) (a) Spherical coordinates; (b) a volume element in spherical coordinates The volume integral of f(r, 0,6) over a spherical region of radius R, centred on the origin, is given by Alternative orderings are also valid. The limits of integration can be adjusted for regions based on parts of a sphere. A thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p= A(r? +y²)2?, where A is a constant. (a) Show that the density expressed in spherical coordinates (r, 0, ø) is p = Ar^(1 – cos? 0) cos? 0. (b) Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral. Hint: You may find the substitution u = cos 0 useful.
Transcribed Image Text:Volume integral over a sphere in spherical coordinates The volume integral of f(r,6,ø) over a spherical region of radius R, centred on the origin, is given by I = . f6r.0.6)r² sin o dr) do) do. I = r sin 0 cos o, y =r sin 0 sin ø, z=r cos 0. 10. Spherical coordinates (r, 0, ø) are related to Cartesian coordinates (x, y, 2) by the equations 1= r sin @ cos o, y=r sin & sin o, 2=r cos 0. In spherical coordinates a volume element has volume 6V = r sin e ốr 0 độ. r sin e (a) (b) (a) Spherical coordinates; (b) a volume element in spherical coordinates The volume integral of f(r, 0,6) over a spherical region of radius R, centred on the origin, is given by Alternative orderings are also valid. The limits of integration can be adjusted for regions based on parts of a sphere. A thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p= A(r? +y²)2?, where A is a constant. (a) Show that the density expressed in spherical coordinates (r, 0, ø) is p = Ar^(1 – cos? 0) cos? 0. (b) Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral. Hint: You may find the substitution u = cos 0 useful.
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