Find the mass of the ball of radius 2 centered at the origin with a density f(p,p,0) = 5e¯p³ The mass is (Type an exact answer, using as needed.)
Find the mass of the ball of radius 2 centered at the origin with a density f(p,p,0) = 5e¯p³ The mass is (Type an exact answer, using as needed.)
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![**Problem Description:**
Find the mass of the ball of radius 2 centered at the origin with a density \( f(\rho, \varphi, \theta) = 5e^{-\rho^3} \).
**Equation:**
The mass is [ ].
(Type an exact answer, using \(\pi\) as needed.)
**Explanation for Solving:**
To find the mass, we need to integrate the given density function over the volume of the ball. The function is given in spherical coordinates:
- \(\rho\) is the radial distance from the origin.
- \(\varphi\) is the polar angle.
- \(\theta\) is the azimuthal angle.
The mass \( M \) of the ball can be calculated using the triple integral in spherical coordinates:
\[
M = \int \int \int_V f(\rho, \varphi, \theta) \cdot \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta
\]
Where the integration limits are:
- \(0 \leq \rho \leq 2\) (radius of the ball),
- \(0 \leq \varphi \leq \pi\),
- \(0 \leq \theta \leq 2\pi\).
The density function is \( f(\rho, \varphi, \theta) = 5e^{-\rho^3} \). Plug this into the integration and solve to find the mass.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c944127-f965-40a6-9c0b-c6a92e7a1d8e%2F3100d09b-6ec7-4772-a24a-e77c235f5904%2Fz83wnj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Find the mass of the ball of radius 2 centered at the origin with a density \( f(\rho, \varphi, \theta) = 5e^{-\rho^3} \).
**Equation:**
The mass is [ ].
(Type an exact answer, using \(\pi\) as needed.)
**Explanation for Solving:**
To find the mass, we need to integrate the given density function over the volume of the ball. The function is given in spherical coordinates:
- \(\rho\) is the radial distance from the origin.
- \(\varphi\) is the polar angle.
- \(\theta\) is the azimuthal angle.
The mass \( M \) of the ball can be calculated using the triple integral in spherical coordinates:
\[
M = \int \int \int_V f(\rho, \varphi, \theta) \cdot \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta
\]
Where the integration limits are:
- \(0 \leq \rho \leq 2\) (radius of the ball),
- \(0 \leq \varphi \leq \pi\),
- \(0 \leq \theta \leq 2\pi\).
The density function is \( f(\rho, \varphi, \theta) = 5e^{-\rho^3} \). Plug this into the integration and solve to find the mass.
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