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Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by
y
=
x
2
,
y
=
0
, and
x
=
5
,
about the
y
-axis.

V
=

Sagot :

Space

Answer:

[tex]\displaystyle V = \frac{625 \pi}{2}[/tex]

General Formulas and Concepts:

Algebra I

  • Functions
  • Function Notation
  • Graphing

Calculus

Integrals

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

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

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

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

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

  • [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:

Step 1: Define

y = x²

y = 0

x = 5

Step 2: Identify

Find other information from graph.

See Attachment.

Bounds of Integration: [0, 5]

Step 3: Find Volume

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

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

Unit: Applications of Integration

Book: College Calculus 10e

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