Mathematics: 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 = 5 , about the y -axis. V =

Please help NO LINKS Use cylindrical shells to find

Ответ:

kayliebug2003

01.08.2021

Mathematics

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

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 = 5

Step 2: Identify

Find other information from graph.

See Attachment.

Bounds of Integration: [0, 5]

Step 3: Find Volume

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

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

Unit: Applications of Integration

Book: College Calculus 10e

Please help NO LINKS

Use cylindrical shells to find the volume of the solid obtained by rotating th

0,0(0 оценок)

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
=
5
,
about the
y
-axis.

V
=

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