Question

Use cylindrical coordinates to evaluate the triple integral dV= function equation editor , where EE is the region that lies inside the cylinder x2+y2=4x2+y2=4 and between the planes z=2z=2 and z=7z=7.

Answer 1

Answer:

$\displaystyle \iiint_E \, dV = 20 \pi$

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]:
$\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Multivariable Calculus

Cylindrical Coordinate Conversions:

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

Volume Formula [Cylindrical Coordinates]:
$\displaystyle V = \iiint_T \, dV \rightarrow V = \iiint_T {r} \, dz \, dr \, d\theta$

Step-by-step explanation:

Step 1: Define

Identify given.

$\displaystyle \text{Region} \ E \left\{ \begin{array}{ccc} \text{Cylinder} \ x^2 + y^2 = 4 \\ \text{Plane} \ z = 2 \\ \text{Plane} \ z = 7 \end{array}$

Step 2: Integrate Pt. 1

Find z bounds.

1. [Given] Define:
   $\displaystyle 2 \leq z \leq 7$

Find r bounds.

1. [Cylinder] Substitute in cylindrical conversions:
   $\displaystyle x^2 + y^2 = 4 \rightarrow r^2 = 4$
2. Simplify:
   $\displaystyle r = \pm 2$
3. [r] Identify:
   $\displaystyle r = 2$
4. Define limits:
   $\displaystyle 0 \leq r \leq 2$

Find θ bounds.

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

Step 3: Integrate Pt. 2

1. [Integrals] Convert [Volume Formula - Cylindrical Coordinates]:
   $\displaystyle \iiint_T \, dV = \iiint_T {r} \, dz \, dr \, d\theta$
2. [Integrals] Substitute in region T:
   $\displaystyle \iiint_T \, dV = \int\limits^{2 \pi}_0 \int\limits^{2}_0 \int\limits^7_2 {r} \, dz \, dr \, d\theta$
3. [dz Integral] Apply Integration Rule [Reverse Power Rule]:
   $\displaystyle \iiint_T \, dV = \int\limits^{2 \pi}_0 \int\limits^{2}_0 {rz \bigg| \limits^{z = 7}_{z = 2}} \, dr \, d\theta$
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus]:
   $\displaystyle \iiint_T \, dV = \int\limits^{2 \pi}_0 \int\limits^{2}_0 {5r} \, dr \, d\theta$
5. [Integrals] Rewrite [Integration Property - Multiplied Constant]:             $\displaystyle \iiint_T \, dV = 5 \int\limits^{2 \pi}_0 \int\limits^{2}_0 {r} \, dr \, d\theta$
6. [dr Integral] Apply Integration Rule [Reverse Power Rule]:
   $\displaystyle \iiint_T \, dV = 5 \int\limits^{2 \pi}_0 {\frac{r^2}{2} \bigg| \limits^{r = 2}_{r = 0}} \, d\theta$
7. Evaluate [Integration Rule - Fundamental Theorem of Calculus]:
   $\displaystyle \iiint_T \, dV = 5 \int\limits^{2 \pi}_0 {2} \, d\theta$
8. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   $\displaystyle \iiint_T \, dV = 10 \int\limits^{2 \pi}_0 {} \, d\theta$
9. [Integral] Apply Integration Rule [Reverse Power Rule]:
   $\displaystyle \iiint_T \, dV = 10 \theta \bigg| \limits^{\theta = 2 \pi}_{\theta = 0}$
10. Evaluate [Integration Rule - Fundamental Theorem of Calculus]:
    $\displaystyle \iiint_T \, dV = 10(2 \pi)$
11. Simplify:
    $\displaystyle \iiint_E \, dV = 20 \pi$

∴ the integral bound by region D is equal to 20π.

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Learn more about cylindrical coordinates: https://brainly.com/question/6177409

Learn more about multivariable calculus: https://brainly.com/question/17203772

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Topic: Multivariable Calculus

Unit: Triple Integrals Applications