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.
The answer would be 0 if I'm correct.
Answer:
m∠y=50°
m∠x=50°
Step-by-step explanation:
we know that
The angles in matching corners are called <u><em>corresponding angles</em></u>. When the two lines are parallel Corresponding Angles are equal
so
m∠y=50° -----> by corresponding angles
and
The angles that are formed on opposite sides of the transversal and inside the two lines are <u><em>alternate interior angles</em></u>. When the lines are parallel, the alternate interior angles are equal.
so
m∠x=m∠y -----> by alternate interior angles
so
m∠x=50°
therefore
m∠y=50°
m∠x=50°
The answer is 8 Feet i think