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lisov135 [29]
2 years ago
14

Please help NO LINKS

Mathematics
1 answer:
Natalka [10]2 years ago
8 0

Answer:

\displaystyle V = \frac{625 \pi}{2}

General Formulas and Concepts:

<u>Algebra I</u>

  • Functions
  • Function Notation
  • Graphing

<u>Calculus</u>

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

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

  • [Shell Method] 2πx is the circumference
  • [Shell Method] 2πxf(x) is the surface area
  • [Shell Method] 2πxf(x)dx is volume

Step-by-step explanation:

<u>Step 1: Define</u>

y = x²

y = 0

x = 5

<u>Step 2: Identify</u>

<em>Find other information from graph.</em>

<em>See Attachment.</em>

Bounds of Integration: [0, 5]

<u>Step 3: Find Volume</u>

  1. Substitute in variables [Shell Method]:                                                         \displaystyle V = 2\pi \int\limits^5_0 {x(x^2)} \, dx
  2. [Integrand] Multiply:                                                                                       \displaystyle V = 2\pi \int\limits^5_0 {x^3} \, dx
  3. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                     \displaystyle V = 2\pi \bigg( \frac{x^4}{4} \bigg) \bigg| \limits^5_0
  4. Evaluate [Integration Rule - FTC 1]:                                                             \displaystyle V = 2\pi \bigg( \frac{625}{4} \bigg)
  5. Multiply:                                                                                                         \displaystyle V = \frac{625 \pi}{2}

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

Unit: Applications of Integration

Book: College Calculus 10e

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