Respuesta :
Answer:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \frac{81 \pi}{10}[/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]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle \text{Region} \ E \left \{ {{\text{Paraboloid:} \ z = 9 - 4(x^2 + y^2)} \atop {\text{Plane:} \ xy}} \right.[/tex]
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV[/tex]
Step 2: Integrate Pt. 1
Find r bounds.
- [Paraboloid] Substitute in xy-plane:
[tex]\displaystyle 0 = 9 - 4(x^2 + y^2)[/tex] - Substitute in cylindrical coordinate conversions:
[tex]\displaystyle 0 = 9 - 4r^2[/tex] - Rewrite:
[tex]\displaystyle r^2 = \frac{9}{4}[/tex] - Solve:
[tex]\displaystyle r = \pm \frac{3}{2}[/tex] - [r] Identify:
[tex]\displaystyle r = \frac{3}{2}[/tex] - Define limits:
[tex]\displaystyle 0 \leq r \leq \frac{3}{2}[/tex]
Find z bounds.
- [Paraboloid] Substitute in cylindrical coordinate conversions:
[tex]\displaystyle z = 9 - 4r^2[/tex] - Define limits:
[tex]\displaystyle 0 \leq z \leq 9 - 4r^2[/tex]
Find θ bounds.
- [Paraboloid] Substitute in xy-plane:
[tex]\displaystyle 0 = 9 - 4(x^2 + y^2)[/tex] - Rewrite:
[tex]\displaystyle x^2 + y^2 = \frac{9}{4}[/tex] - Graph circle [See 2nd attachment]
- [Graph] Identify limits:
[tex]\displaystyle 0 \leq \theta \leq 2 \pi[/tex]
Step 2: Integrate Pt. 2
- [Integrals] Convert [Integral Conversion - Cylindrical Coordinates]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \iiint_E {r\sqrt{x^2 + y^2}} \, dz \, dr \, d\theta[/tex] - [Integrals] Substitute in cylindrical coordinate conversions:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \iiint_E {r\sqrt{r^2}} \, dz \, dr \, d\theta[/tex] - [dz Integrand] Simplify:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \iiint_E {r^2} \, dz \, dr \, d\theta[/tex] - [Integrals] Substitute in region E:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \int\limits^{2 \pi}_0 \int\limits^{\frac{3}{2}}_0 \int\limits^{9 - 4r^2}_0 {r^2} \, dz \, dr \, d\theta[/tex] - [dz Integral] Apply Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \int\limits^{2 \pi}_0 \int\limits^{\frac{3}{2}}_0 {r^2z \bigg| \limits^{9 - 4r^2}_0} \, dr \, d\theta[/tex] - Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \int\limits^{2 \pi}_0 \int\limits^{\frac{3}{2}}_0 {r^2 \big( 9 - 4r^2 \big) } \, dr \, d\theta[/tex] - [dr Integrals] Apply Integration Rules and Properties:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \int\limits^{2 \pi}_0 \Bigg[ 9 \bigg( \frac{r^3}{3} \bigg) - 4 \bigg( \frac{r^5}{5} \bigg) \Bigg] \bigg| \limits^{r = \frac{3}{2}}_{r = 0} \, d\theta[/tex] - Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \int\limits^{2 \pi}_0 {\frac{81}{20}} \, d\theta[/tex] - [Integral] Rewrite [Integration Property - Multiplied Constant]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \frac{81}{20} \int\limits^{2 \pi}_0 {} \, d\theta[/tex] - [Integral] Apply Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \frac{81}{20} \theta \bigg| \limits^{\theta = 2 \pi}_{\theta = 0}[/tex] - Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \iiint_E {\sqrt{x^2 + y^2}} \, dV = \frac{81 \pi}{10}[/tex]
∴ the given triple integral is equal to [tex]\displaystyle \bold{\frac{81 \pi}{10}}[/tex].
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Learn more about cylindrical coordinates: https://brainly.com/question/15223418
Learn more about multivariable calculus: https://brainly.com/question/12269640
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Topic: Multivariable Calculus
Unit: Triple Integrals Applications
