Answer:
[tex]\displaystyle V = \frac{6561 \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 = 9
Step 2: Identify
Find other information from graph.
See attachment.
Bounds of Integration: [0, 9]
Step 3: Find Volume
- Substitute in variables [Shell Method]: [tex]\displaystyle V = 2\pi \int\limits^9_0 {x(x^2)} \, dx[/tex]
- [Integrand] Multiply: [tex]\displaystyle V = 2\pi \int\limits^9_0 {x^3} \, dx[/tex]
- [Integral] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle V = 2\pi \bigg( \frac{x^4}{4} \bigg) \bigg| \limits^9_0[/tex]
- Evaluate [Integration Rule - FTC 1]: [tex]\displaystyle V = 2\pi \bigg( \frac{6561}{4} \bigg)[/tex]
- Multiply: [tex]\displaystyle V = \frac{6561 \pi}{2}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Applications of Integration
Book: College Calculus 10e
