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Find the mass of the solid paraboloid Dequals=​{(r,thetaθ​,z): 0less than or equals≤zless than or equals≤8181minus−r2​, 0less than or equals≤rless than or equals≤99​} with density rhorho​(r,thetaθ​,z)equals=1plus+StartFraction z Over 81 EndFraction z 81. Set up the triple integral using cylindrical coordinates that should be used to find the mass of the solid paraboloid as efficiently as possible. Use increasing limits of integration.

Respuesta :

Space

Answer:

[tex]\displaystyle \int\limits^{2 \pi}_0 \int\limits^{9}_0 \int\limits^{81 - r^2}_0 {r \rho (r, \theta, z)} \, dz \, dr \, d\theta = 4374 \pi[/tex]

General Formulas and Concepts:
Calculus

Integration

  • Integrals

Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Multivariable Calculus

Triple Integration

Cylindrical Coordinate Conversions:

  • [tex]\displaystyle x = r \cos \theta[/tex]
  • [tex]\displaystyle y = r \sin \theta[/tex]
  • [tex]\displaystyle z = z[/tex]
  • [tex]\displaystyle r^2 = x^2 + y^2[/tex]
  • [tex]\displaystyle \tan \theta = \frac{y}{x}}[/tex]

Integral Conversion [Cylindrical Coordinates]:
[tex]\displaystyle \iiint_T \, dV = \iiint_T {r} \, dz \, dr \, d\theta[/tex]

Mass Formula:
[tex]\displaystyle M = \iiint_D {\delta} \, dV[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \rho (r, \theta, z) = 1 + \frac{z}{81}[/tex]

[tex]\displaystyle D = \{ (r, \theta, z): 0 \leq r \leq 9, 0 \leq z \leq 81 - r^2 \}[/tex]

Step 2: Find Mass

  1. [Mass Formula] Convert [Integral Conversion - Cylindrical Coordinates]:
    [tex]\displaystyle M = \iiint_D {\delta} \, dV \rightarrow M = \iiint_D {r \rho} \, dz \, dr \, d\theta[/tex]
  2. [Integrals] Substitute in variables [ρ and region D]:
    [tex]\displaystyle M = \int\limits^{2 \pi}_0 \int\limits^9_0 \int\limits^{81 - r^2}_0 {r \bigg( 1 + \frac{z}{81} \bigg) } \, dz \, dr \, d\theta[/tex]
  3. [dz Integral] Integrate [Integration Rules and Properties]:
    [tex]\displaystyle M = \int\limits^{2 \pi}_0 \int\limits^9_0 { \bigg( zr + \frac{z^2r}{162} \bigg) \bigg| \limits^{z = 81 - r^2}_{z = 0}} \, dr \, d\theta[/tex]
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
    [tex]\displaystyle M = \int\limits^{2 \pi}_0 \int\limits^9_0 { \bigg[ \frac{r \big( r^2 - 243 \big) \big( r^2 - 81 \big) }{162} \bigg] } \, dr \, d\theta[/tex]
  5. [dr Integral] Integrate [Integration Rules and Properties]:
    [tex]\displaystyle M = \int\limits^{2 \pi}_0 { \bigg[ \frac{r^2 \big( r^4 - 486r^2 + 59049 \big)}{972} \bigg] \bigg| \limits^{r = 9}_{r = 0}} \, d\theta[/tex]
  6. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
    [tex]\displaystyle M = \int\limits^{2 \pi}_0 {2187} \, d\theta[/tex]
  7. [Integral] Integrate [Integration Rules and Properties]:
    [tex]\displaystyle M = 2187 \theta \bigg| \limits^{\theta = 2 \pi}_{\theta = 0}[/tex]
  8. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
    [tex]\displaystyle M = 4374 \pi[/tex]

∴ the mass of the solid paraboloid D is equal to 4374π.

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Learn more about cylindrical coordinates: https://brainly.com/question/4110083

Learn more about multivariable calculus: https://brainly.com/question/17203772

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Topic: Multivariable Calculus

Unit: Triple Integrals Applications

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