Answer:
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV = \frac{\pi}{30}[/tex]
General Formulas and Concepts:
Calculus
Integration
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 = 1 - 4(x^2 + y^2)} \atop {\text{Plane:} \ xy}} \right.[/tex][tex]\displaystyle 0 = 1 - 4(x^2 + y^2)[/tex]
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV[/tex]
Step 2: Integrate Pt. 1
Find r bounds.
- [Paraboloid] Substitute in plane xy:
[tex]\displaystyle 0 = 1 - 4(x^2 + y^2)[/tex] - Substitute in cylindrical conversions:
[tex]\displaystyle 0 = 1 - 4r^2[/tex] - Rewrite:
[tex]\displaystyle r^2 = \frac{1}{4}[/tex] - Solve:
[tex]\displaystyle r = \pm \frac{1}{2}[/tex] - [r] Identify:
[tex]\displaystyle r = \frac{1}{2}[/tex] - Define limits:
[tex]\displaystyle 0 \leq r \leq \frac{1}{2}[/tex]
Find z bounds.
- [Paraboloid] Substitute in cylindrical conversions:
[tex]\displaystyle z = 1 - 4r^2[/tex] - Define limits:
[tex]\displaystyle 0 \leq z \leq 1 - 4r^2[/tex]
Find θ bounds.
- [Paraboloid] Substitute in plane xy:
[tex]\displaystyle 0 = 1 - 4(x^2 + y^2)[/tex] - Rewrite:
[tex]\displaystyle x^2 + y^2 = \frac{1}{4}[/tex] - [Circle] Graph [See 2nd Attachment]
- Identify limits:
[tex]\displaystyle 0 \leq \theta \leq 2 \pi[/tex]
Step 3: 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 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{1}{2}}_0 \int\limits^{1 - 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{1}{2}}_0 {r^2z \bigg| \limits^{z = 1 - 4r^2}_{z = 0}} \, dr \, d\theta[/tex] - Evaluate [Integration Rule - FTC 1]:
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV = \int\limits^{2 \pi}_0 \int\limits^{\frac{1}{2}}_0 {\big( r^2 - 4r^4 \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[ \frac{r^3}{3} - 4 \bigg( \frac{r^5}{5} \bigg) \bigg] \bigg| \limits^{r = \frac{1}{2}}_{r = 0} } \, d\theta[/tex] - Evaluate [Integration Rule - FTC 1]:
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV = \int\limits^{2 \pi}_0 {\frac{1}{60}} \, d\theta[/tex] - [Integral] Rewrite [Integration Property - Multiplied Constant]:
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV = \frac{1}{60} \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{1}{60} \theta \bigg| \limits^{\theta = 2 \pi}_{\theta = 0}[/tex] - Evaluate [Integration Rule - FTC 1]:
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV = \frac{1}{60} \big( 2 \pi \big)[/tex] - Simplify:
[tex]\displaystyle \iiint_E {\sqrt{x^2+y^2}} \, dV = \frac{\pi}{30}[/tex]
∴ the given triple integral is equal to [tex]\displaystyle \bold{\frac{\pi}{30}}[/tex].
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Learn more about cylindrical coordinates: https://brainly.com/question/15224796
Learn more about multivariable calculus: https://brainly.com/question/12269640
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Topic: Multivariable Calculus
Unit: Triple Integrals Applications
