Question

Use cylindrical coordinates. evaluate x2 + y2 dv, e where e is the region that lies inside the cylinder x2 + y2 = 25 and between the planes z = −3 and z = 5.

Answer 1

Answer:

$\displaystyle \iiint_E {x^2 + y^2} \, dV = 2500 \pi$

General Formulas and Concepts:
Calculus

Integration

• Integrals

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

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

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

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

Multivariable Calculus

Triple Integrals

Cylindrical Coordinate Conversions:

• $\displaystyle x = r \cos \theta$
• $\displaystyle y = r \sin \theta$
• $\displaystyle z = z$
• $\displaystyle r^2 = x^2 + y^2$
• $\displaystyle \tan \theta = \frac{y}{x}}$

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

Step-by-step explanation:

Step 1: Define

Identify given.

$\displaystyle \iiint_E {x^2 + y^2} \, dV$

$\displaystyle \text{Region} \ E \left \{ {\text{Cylinder:} \ x^2 + y^2 = 25} \atop {\text{Planes:} \ z = -3, 5} \right.$

Step 2: Integrate Pt. 1

Find r bounds.

1. [Cylinder] Substitute in Cylindrical Coordinate Conversions:
   $\displaystyle x^2 + y^2 = 25 \rightarrow r^2 = 25$
2. Solve for r:
   $\displaystyle r = \pm 5$
3. [r] Identify:
   $\displaystyle r = 5$
4. Define limits:
   $\displaystyle 0 \leq r \leq 5$

Find θ bounds.

1. [Cylinder] Graph Circle [See 2nd Attachment]
2. [Graph] Identify limits [Unit Circle]:
   $\displaystyle 0 \leq \theta \leq 2 \pi$

Find z bounds.

1. [Region E] Define limits:
   $\displaystyle -3 \leq z \leq 5$

Step 3: Integrate Pt. 2

1. [Integrals] Convert [Integral Conversion - Cylindrical Coordinates]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \iiint_E {r(x^2 + y^2)} \, dz \, dr \, d\theta$
2. [Integrals] Substitute in Cylindrical Coordinate Conversions:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \iiint_E {r^3} \, dz \, dr \, d\theta$
3. [Integrals] Substitute in region E:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \int\limits^{2 \pi}_0 \int\limits^5_0 \int\limits^5_{-3} {r^3} \, dz \, dr \, d\theta$
4. [dz Integral] Apply Integration Rule [Reverse Power Rule]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \int\limits^{2 \pi}_0 \int\limits^5_0 {zr^3 \bigg| \limits^{z = 5}_{z = -3}} \, dr \, d\theta$
5. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \int\limits^{2 \pi}_0 \int\limits^5_0 {8r^3} \, dr \, d\theta$
6. [dr Integral] Apply Integration Rules and Properties [Reverse Power Rule + Multiplied Constant]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \int\limits^{2 \pi}_0 {2r^4 \bigg| \limits^{r = 5}_{r = 0}} \, d\theta$
7. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = \int\limits^{2 \pi}_0 {1250} \, d\theta$
8. [Integral] Apply Integration Rules and Properties [Reverse Power Rule + Multiplied Constant]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = 1250 \theta \bigg| \limits^{\theta = 2 \pi}_{\theta = 0}$
9. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   $\displaystyle \iiint_E {x^2 + y^2} \, dV = 2500 \pi$

∴ the given integral using cylindrical coordinates equals 2500π.

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Learn more about cylindrical coordinates: https://brainly.com/question/7662645

Learn more about multivariable calculus: https://brainly.com/question/9381576
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Topic: Multivariable Calculus

Unit: Triple Integrals Applications