Question

Use cylindrical coordinates. Find the volume of the solid that lies within both the cylinder x2 + y2 = 25 and the sphere x2 + y2 + z2 = 49.

Answer 1

Answer:

Step-by-step explanation:

we are asked to find the volume of solid that lies within both the cylinder

$x^2 + y^2 = 25$

and

the sphere$x^2 + y^2 + z^2 = 49.$

Conversion from rectangular to cylindrical is

$x=rcost\\y = rsint\\z=z$

|J| =r

In cylindrical coordinates the volume is bounded by the cylinder r=5 and

$r^2+z^2 =49$

Hence we can write volume as

$\int \int \int dxdydz\\=\\\int _0^5 \int_0^{2\pi} \int_{-\sqrt{49-r^2} } ^{\sqrt{49-r^2} rdzdtdr\\= 2\pi \int _0^5 (2\sqrt{49-r^2} rdr\\=4\pi (-(49-r^2) (2/3)\\= \frac{4\pi}{3} (343-48\sqrt{6} )$

Answer 2

Answer:

$\displaystyle \iiint_T \, dV = \frac{4 \pi \big(343 - 48 \sqrt{6} \big) }{3}$

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]:
$\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

Integration Method: U-Substitution

Multivariable Calculus

Triple Integrals

Cylindrical Coordinate Conversions:

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

Volume Formula:
$\displaystyle V = \iiint_D \, dV$

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.

$\displaystyle \text{Region} \ T \left \{ {{\text{Cylinder:} \ x^2 + y^2 = 25} \atop {\text{Sphere:} \ x^2 + y^2 + z^2 = 49}} \right.$

Step 2: Find Volume Pt. 1

Find z bounds.

1. [Sphere] Substitute in cylindrical coordinate conversions:
   $\displaystyle r^2 + z^2 = 49$
2. Solve for z:
   $\displaystyle z = \pm \sqrt{49 - r^2}$
3. Define limits:
   $\displaystyle - \sqrt{49 - r^2} \leq z \leq \sqrt{49 - r^2}$

Find θ bounds.

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

Find r bounds.

1. [Circle] Substitute in cylindrical coordinate conversions:
   $\displaystyle r^2 = 25$
2. [r] Solve:
   $\displaystyle r = \pm 5$
3. [r] Identify:
   $\displaystyle r = 5$
4. Define limits:
   $\displaystyle 0 \leq r \leq 5$

Step 3: Find Volume Pt. 2

1. [Volume Formula] Convert [Volume Formula - Cylindrical Coordinates]:
   $\displaystyle V = \iiint_T {r} \, dz \, dr \, d\theta$
2. [Integrals] Substitute in region T:
   $\displaystyle V = \int\limits^{2 \pi}_0 \int\limits^5_0 \int\limits^{\sqrt{49 - r^2}}_{- \sqrt{49 - r^2}} {r} \, dz \, dr \, d\theta$
3. [dz Integral] Integrate [Integration Rule - Reverse Power Rule]:
   $\displaystyle V = \int\limits^{2 \pi}_0 \int\limits^5_0 {rz \bigg| \limits^{z = \sqrt{49 - r^2}}_{z = - \sqrt{49 - r^2}}} \, dr \, d\theta$
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   $\displaystyle V = \int\limits^{2 \pi}_0 \int\limits^5_0 {2r\sqrt{49-r^2}} \, dr \, d\theta$
5. [dr Integral] Integrate [Integration Rules, Properties, and Methods]:
   $\displaystyle V = \int\limits^{2 \pi}_0 {\frac{-2 \big( 49 - r^2 \big) ^\bigg{\frac{3}{2}} }{3} \bigg| \limits^{r = 5}_{r = 0}} \, d\theta$
6. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   $\displaystyle V = \int\limits^{2 \pi}_0 {\frac{-2 \big( 48 \sqrt{6} - 343 \big) }{3}} \, d\theta$
7. [Integral] Integrate [Integration Rule - Reverse Power Rule]:
   $\displaystyle V = \frac{-2 \big( 48 \sqrt{6} - 343 \big) }{3} \theta \bigg| \limits^{\theta = 2 \pi}_{\theta = 0}$
8. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   $\displaystyle V = \frac{4 \pi \big(343 - 48 \sqrt{6} \big) }{3}$

∴ the volume of the solid that lies between the regions using cylindrical coordinates is approximately 944.256.

---

Learn more about cylindrical coordinates: https://brainly.com/question/15579112

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

Topic: Multivariable Calculus

Unit: Triple Integrals Applications