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klio [65]
3 years ago
9

Use cylindrical shells to find the volume of the solid generated when the region

Mathematics
1 answer:
Kisachek [45]3 years ago
6 0

Answer:

\displaystyle V = \frac{176 \pi}{15}

General Formulas and Concepts:

<u>Pre-Algebra</u>

  • Equality Properties

<u>Algebra I</u>

  • Terms/Coefficients
  • Expanding
  • Functions
  • Function Notation
  • Graphing
  • Exponential Rule [Root Rewrite]:                                                                     \displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}

<u>Calculus</u>

Integrals

  • Definite Integrals
  • Area under the curve

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

Shell Method:

\displaystyle V = 2\pi \int\limits^b_a {xf(x)} \, dx

  • [Shell Method] <em>x</em> is the radius
  • [Shell Method] 2πx is the circumference
  • [Shell Method] 2πxf(x) is the surface area
  • [Shell Method] 2πxf(x)dx is the volume

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

Graph of region

y = x²

x = 2

y = 4

Axis of Revolution: y = -1

<u>Step 2: Sort</u>

<em>We are revolving around a horizontal line.</em>

  1. [Function] Rewrite in terms of <em>y</em>:                                                                      x = √y
  2. [Graph] Identify bounds of integration:                                                           [0, 4]

<u>Step 3: Find Volume Pt. 1</u>

  1. [Shell Method] Find distance of radius <em>x</em>:                                                       x = y + 1
  2. [Shell Method] Find circumference variable f(x) [Area]:                                 \displaystyle f(x) = 2 - \sqrt{y}
  3. [Shell Method] Substitute in variables:                                                           \displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - \sqrt{y})} \, dy
  4. [Integral] Rewrite integrand [Exponential Rule - Root Rewrite]:                    \displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - y^\bigg{\frac{1}{2}})} \, dy
  5. [Integral] Expand integrand:                                                                            \displaystyle V = 2\pi \int\limits^4_0 {(-y^\bigg{\frac{3}{2}} + 2y - y^\bigg{\frac{1}{2}} + 2)} \, dy
  6. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                        \displaystyle V = 2\pi \bigg( \frac{-2y^\bigg{\frac{5}{2}}}{5} + y^2 - \frac{2y^\bigg{\frac{3}{2}}}{3} + 2y \bigg) \bigg| \limits^4_0
  7. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:              \displaystyle V = 2\pi (\frac{88}{15})
  8. Multiply:                                                                                                             \displaystyle V = \frac{176 \pi}{15}

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Applications of Integration

Book: College Calculus 10e

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