Question

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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
=
9
,
about the
y
-axis.

V
=

Answer 1

`\displaystyle V = (6561 \pi)/(2)`

General Formulas and Concepts:

Algebra I

• Functions
• Function Notation
• Graphing

Calculus

Integrals

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (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

Explanation:

Step 1: Define

y = x²

y = 0

x = 9

Step 2: Identify

Find other information from graph.

See attachment.

Bounds of Integration: [0, 9]

Step 3: Find Volume

1. Substitute in variables [Shell Method]:
   `\displaystyle V = 2\pi \int\limits^9_0 {x(x^2)} \, dx`
2. [Integrand] Multiply:
   `\displaystyle V = 2\pi \int\limits^9_0 {x^3} \, dx`
3. [Integral] Integrate [Integration Rule - Reverse Power Rule]:
   `\displaystyle V = 2\pi \bigg( (x^4)/(4) \bigg) \bigg| \limits^9_0`
4. Evaluate [Integration Rule - FTC 1]:
   `\displaystyle V = 2\pi \bigg( (6561)/(4) \bigg)`
5. Multiply:
   `\displaystyle V = (6561 \pi)/(2)`

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

Unit: Applications of Integration

Book: College Calculus 10e