Let D be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere. Express the volume of D as an iterated triple integral in spherica... Let D be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere. Express the volume of D as an iterated triple integral in spherical coordinates.
Understand the Problem
The question is asking to represent the volume of a spherical cap cut from a solid ball in spherical coordinates using a triple integral. This involves understanding the geometry of the sphere and how to set up the integral limits based on the given parameters.
Answer
The volume of the cap \( D \) is given by $$ V = \int_0^{2\pi} \int_0^{\frac{\pi}{3}} \int_0^2 \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta $$
Answer for screen readers
The volume of the cap ( D ) is expressed as the iterated triple integral:
$$ V = \int_0^{2\pi} \int_0^{\frac{\pi}{3}} \int_0^2 \rho^2 \sin(\phi) , d\rho , d\phi , d\theta $$
Steps to Solve
-
Understand the Geometry of the Sphere The sphere has a radius of 2 units. The plane that cuts the sphere is located 1 unit away from the center, which means it is positioned at ( z = 1 ).
-
Determine the Limits for ( z ) in Spherical Coordinates In spherical coordinates, ( z ) is represented as ( z = \rho \cos(\phi) ), where ( \rho ) is the radial distance and ( \phi ) is the polar angle. The cap volume is from the center of the sphere up to the plane:
- The lower limit for ( z ): ( z = 0 ) (the bottom of the cap)
- The upper limit for ( z ): ( z = 1 ) (the plane location). Thus, we set ( 1 = \rho \cos(\phi) ), giving us the equation for the upper limit.
-
Set Limits for ( \rho ) The radius of the sphere establishes the radial limit:
- The limit for ( \rho ): ( 0 \leq \rho \leq 2 ).
-
Determine Limits for ( \phi ) The angle ( \phi ) ranges from 0 (positive ( z )-axis) to an angle defined by where the plane intersects the sphere, which can be found using: $$ \cos(\phi) = \frac{1}{2} $$ Therefore, ( \phi ) will range from ( 0 ) to ( \frac{\pi}{3} ).
-
Determine Limits for ( \theta ) The angle ( \theta ) can wrap around the entire sphere:
- The limit for ( \theta ): ( 0 \leq \theta < 2\pi ).
-
Set Up the Integral The volume ( V ) of the spherical cap ( D ) can be expressed as: $$ V = \int_0^{2\pi} \int_0^{\frac{\pi}{3}} \int_0^2 \rho^2 \sin(\phi) , d\rho , d\phi , d\theta $$
The volume of the cap ( D ) is expressed as the iterated triple integral:
$$ V = \int_0^{2\pi} \int_0^{\frac{\pi}{3}} \int_0^2 \rho^2 \sin(\phi) , d\rho , d\phi , d\theta $$
More Information
This integral represents the volume of a spherical cap cut from a solid sphere. The spherical coordinates help simplify the calculations by utilizing the symmetry of the sphere.
Tips
- Misunderstanding the limits for ( z ) based on the physical setup of the problem. It's vital to accurately translate the height where the plane intersects the sphere.
- Forgetting to convert the limits for the angles when switching to spherical coordinates.
AI-generated content may contain errors. Please verify critical information
