Answer:
![A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\approx9.8524](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cfrac%7B16%7D%7B3%7D-2%5Cln%28%7C3%7C%29%5Cbiggr%5D%5Capprox9.8524)
Step-by-step explanation:
Use the Washer Method ![A=\pi\displaystyle \int\limits_{a}^{b}{\bigr[R(x)^2-r(x)^2\bigr] \, dx](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%20%5Cint%5Climits_%7Ba%7D%5E%7Bb%7D%7B%5Cbigr%5BR%28x%29%5E2-r%28x%29%5E2%5Cbigr%5D%20%5C%2C%20dx)
where 
is the outer radius and 
is the inner radius.
If we sketch out the graph, we see that 
intersects points 
and 
, which will be our bounds of integration.
Here, our outer radius will be 
and our inner radius will be 
.
Thus, we can compute the integral and find the volume:
![A=\pi\displaystyle\int\limits^{3}_{1} {(-2)^2-\biggr(-1-\frac{1}{x}\biggr)^2 } \, dx\\ \\A=\pi\displaystyle\int\limits^{3}_{1} {4-\biggr(1+\frac{2}{x}+\frac{1}{x^2} \biggr) } \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {4-1-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {3-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\biggr[3x-2\ln(|x|)+\frac{1}{x}\biggr]\Biggr|_{1}^{3}\\](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B%28-2%29%5E2-%5Cbiggr%28-1-%5Cfrac%7B1%7D%7Bx%7D%5Cbiggr%29%5E2%20%7D%20%5C%2C%20dx%5C%5C%20%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B4-%5Cbiggr%281%2B%5Cfrac%7B2%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%5E2%7D%20%20%5Cbiggr%29%20%7D%20%5C%2C%20dx%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B4-1-%5Cfrac%7B2%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%20%5C%2C%20dx%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cint%5Climits%5E%7B3%7D_%7B1%7D%20%7B3-%5Cfrac%7B2%7D%7Bx%7D-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%20%5C%2C%20dx%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B3x-2%5Cln%28%7Cx%7C%29%2B%5Cfrac%7B1%7D%7Bx%7D%5Cbiggr%5D%5CBiggr%7C_%7B1%7D%5E%7B3%7D%5C%5C)
![A=\pi\displaystyle\biggr[\biggr(3(3)-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3(1)-2\ln(|1|)+\frac{1}{1}\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(9-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3+1\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(\frac{28}{3}-2\ln(|3|)\biggr)-\biggr(4\biggr)\biggr]\\A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\\A\approx9.8524](https://tex.z-dn.net/?f=A%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cbiggr%283%283%29-2%5Cln%28%7C3%7C%29%2B%5Cfrac%7B1%7D%7B3%7D%5Cbiggr%29-%5Cbiggr%283%281%29-2%5Cln%28%7C1%7C%29%2B%5Cfrac%7B1%7D%7B1%7D%5Cbiggr%29%5Cbiggr%5D%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cbiggr%289-2%5Cln%28%7C3%7C%29%2B%5Cfrac%7B1%7D%7B3%7D%5Cbiggr%29-%5Cbiggr%283%2B1%5Cbiggr%29%5Cbiggr%5D%5C%5C%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cbiggr%28%5Cfrac%7B28%7D%7B3%7D-2%5Cln%28%7C3%7C%29%5Cbiggr%29-%5Cbiggr%284%5Cbiggr%29%5Cbiggr%5D%5C%5CA%3D%5Cpi%5Cdisplaystyle%5Cbiggr%5B%5Cfrac%7B16%7D%7B3%7D-2%5Cln%28%7C3%7C%29%5Cbiggr%5D%5C%5CA%5Capprox9.8524)
In conclusion, the volume of the solid of revolution will be about 9.8524 cubic units. See the attached graph for a helpful visual!
Answer:
The average of the four numbers is 79.
Answer:
No, it is not a function
Step-by-step explanation:
Using the vertical line test, you would find that it would hit two points, making it not a funtion.
Answer:
just practice time and again practice might not make perfect but it surely makes better. relying when your teacher only might not help you must mostly rely on your hard work and also have people at your disposal to help you including the said teacher hope you'll improve
Answer:
Option A is correct, i.e. 12.
Step-by-step explanation:
Given is the table of x&y relationship which represents an exponential function.
The average rate of change for a function can be found using the following formula:-
F_average = { f(b) - f(a) } / (b-a)
Given a = 3 and b = 5.
From the table, f(3) = 8 and f(5) = 32.
So, F_average = (32-8)/(5-3)
F_average = 24/2 = 12.
Hence, option A is correct, i.e. 12.
