Question

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Answer 1

Answer:

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

General Formulas and Concepts:

Algebra I

• Functions
• Function Notation
• Graphing

Calculus

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:

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]:                                                         $\displaystyle V = 2\pi \int\limits^5_0 {x(x^2)} \, dx$
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