Answer:
![\displaystyle V = \frac{176 \pi}{15}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%20%5Cfrac%7B176%20%5Cpi%7D%7B15%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
<u>Algebra I</u>
- Terms/Coefficients
- Expanding
- Functions
- Function Notation
- Graphing
- Exponential Rule [Root Rewrite]:
![\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%5Bn%5D%7Bx%7D%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
<u>Calculus</u>
Integrals
- Definite Integrals
- Area under the curve
Integration Rule [Reverse Power Rule]: ![\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bx%5En%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bx%5E%7Bn%20%2B%201%7D%7D%7Bn%20%2B%201%7D%20%2B%20C)
Integration Rule [Fundamental Theorem of Calculus 1]: ![\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Eb_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20F%28b%29%20-%20F%28a%29)
Integration Property [Multiplied Constant]: ![\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bcf%28x%29%7D%20%5C%2C%20dx%20%3D%20c%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Integration Property [Addition/Subtraction]: ![\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%5Cpm%20%5Cint%20%7Bg%28x%29%7D%20%5C%2C%20dx)
Shell Method:
![\displaystyle V = 2\pi \int\limits^b_a {xf(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%202%5Cpi%20%5Cint%5Climits%5Eb_a%20%7Bxf%28x%29%7D%20%5C%2C%20dx)
- [Shell Method] <em>x</em> 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:
<u>Step 1: Define</u>
<em>Identify</em>
Graph of region
y = x²
x = 2
y = 4
Axis of Revolution: y = -1
<u>Step 2: Sort</u>
<em>We are revolving around a horizontal line.</em>
- [Function] Rewrite in terms of <em>y</em>: x = √y
- [Graph] Identify bounds of integration: [0, 4]
<u>Step 3: Find Volume Pt. 1</u>
- [Shell Method] Find distance of radius <em>x</em>:
![x = y + 1](https://tex.z-dn.net/?f=x%20%3D%20y%20%2B%201)
- [Shell Method] Find circumference variable f(x) [Area]:
![\displaystyle f(x) = 2 - \sqrt{y}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%29%20%3D%202%20-%20%5Csqrt%7By%7D)
- [Shell Method] Substitute in variables:
![\displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - \sqrt{y})} \, dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%202%5Cpi%20%5Cint%5Climits%5E4_0%20%7B%28y%20%2B%201%29%282%20-%20%5Csqrt%7By%7D%29%7D%20%5C%2C%20dy)
- [Integral] Rewrite integrand [Exponential Rule - Root Rewrite]:
![\displaystyle V = 2\pi \int\limits^4_0 {(y + 1)(2 - y^\bigg{\frac{1}{2}})} \, dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%202%5Cpi%20%5Cint%5Climits%5E4_0%20%7B%28y%20%2B%201%29%282%20-%20y%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B2%7D%7D%29%7D%20%5C%2C%20dy)
- [Integral] Expand integrand:
![\displaystyle V = 2\pi \int\limits^4_0 {(-y^\bigg{\frac{3}{2}} + 2y - y^\bigg{\frac{1}{2}} + 2)} \, dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%202%5Cpi%20%5Cint%5Climits%5E4_0%20%7B%28-y%5E%5Cbigg%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%2B%202y%20-%20y%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%2B%202%29%7D%20%5C%2C%20dy)
- [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](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%202%5Cpi%20%5Cbigg%28%20%5Cfrac%7B-2y%5E%5Cbigg%7B%5Cfrac%7B5%7D%7B2%7D%7D%7D%7B5%7D%20%2B%20y%5E2%20-%20%5Cfrac%7B2y%5E%5Cbigg%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B3%7D%20%2B%202y%20%5Cbigg%29%20%5Cbigg%7C%20%5Climits%5E4_0)
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
![\displaystyle V = 2\pi (\frac{88}{15})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%202%5Cpi%20%28%5Cfrac%7B88%7D%7B15%7D%29)
- Multiply:
![\displaystyle V = \frac{176 \pi}{15}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20V%20%3D%20%5Cfrac%7B176%20%5Cpi%7D%7B15%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Applications of Integration
Book: College Calculus 10e
The greatest common denominator for 14 and 21 is 7. (14 / 7 = 2 21 / 7 = 3). So, if you had groups of 7 you would have 2 groups of boys and 3 groups of girls which gives you a total of 5 groups.
Answer:
About 32 times more
Step-by-step explanation:
3 * 10^2 = 300
9 * 10^0 = 9 * 1 = 9.
Number of teachers who atten = 300 - 9
= 291.
291 / 9
= 32.333..
<h2>
<u>Answer</u><u>:</u><u>-</u></h2>
Jessica sells = 240 books,
Josh sells = 16 books.
<h3>
Given:-</h3>
The price of one book is = $15
Jessica sold 15 times more books than Josh.
In total, they have sold 256 books.
<h3>Solution:-</h3>
We can easily find the answer using variables.
Let,
Josh sold a total of x books whereas Jessica sold a total number of 15x books.
together they sold 16x books.
So we can form the equation
16x= 256
by solving this equation we will get x=16
So, Josh sold 16 books and Jessica sold 16×15=240 books in total.