Use cylindrical coordinates. Evaluate 9x(x2 + y2) dV, where E is the solid in the first octant that lies beneath the paraboloid z = 4 − x2 − y2.

Answer:[tex]\displaystyle \iiint_E {9x \big( x^2 + y^2 \big)} \, dV = \frac{2304}{35}[/tex]General Formulas and Concepts:
CalculusIntegrationIntegralsIntegration 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 CalculusTriple IntegrationCylindrical 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: DefineIdentify 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. 1Find 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.---Learn more about cylindrical coordinates: more about multivariable calculus: : Multivariable CalculusUnit: Triple Integrals Applications