Answer:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \frac{2304}{35}[/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:
*Note:
It is implied that the region E is also bounded by the xy-plane.
Step 1: Define
Identify given.
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV[/tex]
[tex]\displaystyle \text{Region} \ E \left \{ {{\text{Paraboloid:} \ z = 4 - x^2 - y^2} \atop {\text{Plane:} \ xy}} \right.[/tex]
Step 2: Integrate Pt. 1
Find z bounds.
- [Paraboloid] Rewrite:
[tex]\displaystyle z = 4 - (x^2 + y^2)[/tex] - Substitute in Cylindrical Coordinate Conversions:
[tex]\displaystyle z = 4 - r^2[/tex] - Define limits:
[tex]\displaystyle 0 \leq z \leq 4 - r^2[/tex]
Find r bounds.
- [Cylindrical Paraboloid] Substitute in xy-plane (z = 0):
[tex]\displaystyle 0 = 4 - r^2[/tex] - Solve for r:
[tex]\displaystyle r = \pm 2[/tex] - [r] Identify:
[tex]\displaystyle r = 2[/tex] - Define limits:
[tex]\displaystyle 0 \leq r \leq 2[/tex]
Find θ bounds.
- [Paraboloid] Substitute in xy-plane (z = 0):
[tex]\displaystyle 0 = 4 - (x^2 + y^2)[/tex] - Rewrite:
[tex]\displaystyle x^2 + y^2 = 4[/tex] - [Circle] Graph [See 2nd Attachment]
- [Graph] Identify limits:
[tex]\displaystyle 0 \leq \theta \leq \frac{\pi}{2}[/tex]
Step 3: Integrate Pt. 2
- [Integrals] Convert [Integral Conversion - Cylindrical Coordinates]:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \iiint_E {9x \big( x^2 + y^2 \big)r} \, dz \, dr \, d\theta[/tex] - [dz Integrand] Substitute in Cylindrical Coordinate Conversions:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \iiint_E {9(r \cos \theta)r^2r} \, dz \, dr \, d\theta[/tex] - [dz Integrand] Simplify:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \iiint_E {9r^4 \cos \theta} \, dz \, dr \, d\theta[/tex] - [Integrals] Substitute in region E:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \int\limits^{\frac{\pi}{2}}_0 \int\limits^2_0 \int\limits^{4 - r^2}_0 {9r^4 \cos \theta} \, dz \, dr \, d\theta[/tex] - [dz Integral] Apply Integration Rules and Properties:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \int\limits^{\frac{\pi}{2}}_0 \int\limits^2_0 {9r^4z \cos \theta \bigg| \limits^{z = 4 - r^2}_{z = 0}} \, dr \, d\theta[/tex] - Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \int\limits^{\frac{\pi}{2}}_0 \int\limits^2_0 { \bigg[ 9r^4(r^2 - 4) \cos \theta \bigg] } \, dr \, d\theta[/tex] - [dr Integral] Apply Integration Rules and Properties:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \int\limits^{\frac{\pi}{2}}_0 { \bigg[ \frac{-9r^5)5r^2 - 28) \cos \theta}{35} \bigg] \bigg| \limits^{r = 2}_{r = 0}} \, d\theta[/tex] - Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \int\limits^{\frac{\pi}{2}}_0 { \frac{2304 \cos \theta}{35}} \, d\theta[/tex] - [Integral] Apply Trigonometric Integration [Integration Rules and Properties]:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \frac{2304 \sin \theta}{35}} \bigg| \limits^{\theta = \frac{\pi}{2}}_{\theta = 0}[/tex] - Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \frac{2304}{35}[/tex]
∴ the given integral defined by the region E is approximately 65.8286.
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Learn more about cylindrical coordinates: https://brainly.com/question/14004645
Learn more about multivariable calculus: https://brainly.com/question/17203772
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Topic: Multivariable Calculus
Unit: Triple Integrals Applications
