Mathematics, 08.07.2021 19:10 Karumbafaith

Use cylindrical shells to find the volume of the solid generated when the region R under y = x2 over the interval (0,2) revolved about the line y = -1

Use cylindrical shells to find the volume of the solid generated when the region

R under y = x2 o

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Answer from: 19thomasar

$\displaystyle V = \frac{176 \pi}{15}$

General Formulas and Concepts:

Pre-Algebra

Equality Properties

Algebra I

Terms/CoefficientsExpandingFunctionsFunction NotationGraphingExponential Rule [Root Rewrite]: $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$

Calculus

Integrals

Definite IntegralsArea under the curve

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] x is the radius[Shell Method] 2πx is the circumference[Shell Method] 2πxf(x) is the surface area[Shell Method] 2πxf(x)dx is the volume

Step-by-step explanation:

Step 1: Define

Identify

Graph of region

y = x²

x = 2

y = 4

Axis of Revolution: y = -1

Step 2: Sort

We are revolving around a horizontal line.

[Function] Rewrite in terms of y: x = √y[Graph] Identify bounds of integration: [0, 4]

Step 3: Find Volume Pt. 1

[Shell Method] Find distance of radius x: x = y + 1

[Shell Method] Find circumference variable f(x) [Area]: $\displaystyle f(x) = 2 - \sqrt{y}$ [Shell Method] Substitute in variables: $\displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - \sqrt{y})} \, dy$ [Integral] Rewrite integrand [Exponential Rule - Root Rewrite]: $\displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - y^\bigg{\frac{1}{2}})} \, dy$ [Integral] Expand integrand: $\displaystyle V = 2\pi \int\limits^4_0 {(-y^\bigg{\frac{3}{2}} + 2y - y^\bigg{\frac{1}{2}} + 2)} \, dy$ [Integral] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle V = 2\pi \bigg( \frac{-2y^\bigg{\frac{5}{2}}}{5} + y^2 - \frac{2y^\bigg{\frac{3}{2}}}{3} + 2y \bigg) \bigg| \limits^4_0$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle V = 2\pi (\frac{88}{15})$ Multiply: $\displaystyle V = \frac{176 \pi}{15}$

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

Unit: Applications of Integration

Book: College Calculus 10e

Answer from: Quest

option a.

step-by-step explanation:

if you look closely at the graph, we can notice that the function is not defined at two different points.

if you look at functions b and c, you will notice that both functions are not defined at just one point. the function b is not defined at x= -5/2 and function c is not defined at x=-5. both options are discarded, given that our function should not be defined at two fiferent points.

now, if you look closely at the graph you will notice that the function sketched is not defined at x=5 and x=-2.

Which of the following rational functions is graphed below ?

Answer from: Quest

Peanuts r great aren’t they it just has a good taste

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Use cylindrical shells to find the volume of the solid generated when the region R under y = x2 ove...