When integrating using shells, the first step is to plot the graph. I personally plotted it with the x and y axis switched, because it aids me in picturing the graph. The integral ranges from 9 to 11, since those are the limits from the two lines y = 9 and y = 11. The reason that these are the limits is because it is rotating around the x, not the y.
![\int\limits^{11}_{9}](https://tex.z-dn.net/?f=%20%5Cint%5Climits%5E%7B11%7D_%7B9%7D)
Now that you have the limits of the integral, you have to find what goes inside it.
Because you are integrating using shells, you need to remember to include the
![2\pi y](https://tex.z-dn.net/?f=2%5Cpi%20y)
Again, there is a y here instead of an x, because you are rotating around the x axis. Then you just need to input the function f(y). If you look at the graph that you (hopefully) plotted, you can see that this function ranges between the y axis and the curve
![f(y) = \frac{9}{y}](https://tex.z-dn.net/?f=f%28y%29%20%3D%20%5Cfrac%7B9%7D%7By%7D)
. Put together the pieces, and you have the integral
![\int\limits^{11}_{9} {2 \pi y*f(y)} \, dy](https://tex.z-dn.net/?f=%20%5Cint%5Climits%5E%7B11%7D_%7B9%7D%20%7B2%20%5Cpi%20y%2Af%28y%29%7D%20%5C%2C%20dy%20)
After substituting in
![f(y) = \frac{9}{y}](https://tex.z-dn.net/?f=f%28y%29%20%3D%20%5Cfrac%7B9%7D%7By%7D)
, you get
![\int\limits^{11}_{9} {2 \pi y*\frac{9}{y}} \, dy](https://tex.z-dn.net/?f=%20%5Cint%5Climits%5E%7B11%7D_%7B9%7D%20%7B2%20%5Cpi%20y%2A%5Cfrac%7B9%7D%7By%7D%7D%20%5C%2C%20dy%20)
Simplified, this is
![18\pi\int\limits^{11}_{9} \, dy](https://tex.z-dn.net/?f=%2018%5Cpi%5Cint%5Climits%5E%7B11%7D_%7B9%7D%20%5C%2C%20dy%20)
Integrating, we get
![18\pi * 2](https://tex.z-dn.net/?f=%2018%5Cpi%20%2A%202)
Therefore, the solution is
![36 \pi](https://tex.z-dn.net/?f=%2036%20%5Cpi%20)
Note: I didn't spend very much time reviewing these integrals, so I may be incorrect.
Answer:
2,294.66 square inches
Step-by-step explanation:
The area of an octagon is given by the formula:
A = 2 * a² * ( 1 + √2 )
So, if we input the numbers we have:
A = 2 * 21.8² * ( 1 + √2 )
A = 2 * 475.24 * (2.4142)
A = 2,294.66 square inches
As you can see, if you double the side of an octagon, its area will quadruples, which is logic since the one variable in the calculation of the area is the side's length... that is squared. So, if you double it, that double factor is squared.
It's as if we had written
A = 2 * (10.9 * 2)² * ( 1 + √2 )
A = 2 * (10.9² * 2²) * ( 1 + √2 )
Answer:
The value of f(x) - g(x) is:
f(x) - g(x) = 2x² + 6x - 4
Step-by-step explanation:
Given
f(x) = 3x² + x - 3
g(x) = x² - 5x + 1
To determine
f(x) - g(x)
Using the formula
f(x) - g(x) = 3x² + x - 3 - ( x² - 5x + 1)
= 3x² + x - 3 - x² + 5x - 1
= 2x² + 6x - 4
Therefore, the value of f(x) - g(x) is:
f(x) - g(x) = 2x² + 6x - 4
The answer is..
Yo Face Bruh!!!
Answer:
C) ![x=\frac{\log(6)}{2\log(3)}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%5Clog%286%29%7D%7B2%5Clog%283%29%7D)
Step-by-step explanation:
![15(3)^{2x}=90\\\\3^{2x}=6\\\\2x=\log_3(6)\\\\2x=\frac{\log(6)}{\log(3)}\\\\x=\frac{\log(6)}{2\log(3)}](https://tex.z-dn.net/?f=15%283%29%5E%7B2x%7D%3D90%5C%5C%5C%5C3%5E%7B2x%7D%3D6%5C%5C%5C%5C2x%3D%5Clog_3%286%29%5C%5C%5C%5C2x%3D%5Cfrac%7B%5Clog%286%29%7D%7B%5Clog%283%29%7D%5C%5C%5C%5Cx%3D%5Cfrac%7B%5Clog%286%29%7D%7B2%5Clog%283%29%7D)
Remember to use the change of base formula, which states that
so that everything has the same base of 10