Mathematics, 02.07.2021 01:00 alexmoy45p8yd7v

Please Help NO LINKS 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
=

Answers: 1

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Answer from: ari313

$\displaystyle V = \frac{6561 \pi}{2}$

General Formulas and Concepts:

Algebra I

FunctionsFunction NotationGraphing

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 = 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]: $\displaystyle V = 2\pi \int\limits^9_0 {x(x^2)} \, dx$ [Integrand] Multiply: $\displaystyle V = 2\pi \int\limits^9_0 {x^3} \, dx$ [Integral] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle V = 2\pi \bigg( \frac{x^4}{4} \bigg) \bigg| \limits^9_0$ Evaluate [Integration Rule - FTC 1]: $\displaystyle V = 2\pi \bigg( \frac{6561}{4} \bigg)$ Multiply: $\displaystyle V = \frac{6561 \pi}{2}$

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

Unit: Applications of Integration

Book: College Calculus 10e

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