Yes it's exactly due to symmetry. Specifically, symmetry about the x axis.
Simply writing ![\displaystyle \int_{-8}^{-4}\left[\sqrt{x+8} \ \right]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_%7B-8%7D%5E%7B-4%7D%5Cleft%5B%5Csqrt%7Bx%2B8%7D%20%5C%20%5Cright%5Ddx)
, without the 2 out front, will only get you the area of the portion shown in red (see diagram below).
The blue region has equal area of the red region due to symmetry.
So,
![\displaystyle 2\int_{-8}^{-4}\left[\sqrt{x+8} \ \right]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%5Cint_%7B-8%7D%5E%7B-4%7D%5Cleft%5B%5Csqrt%7Bx%2B8%7D%20%5C%20%5Cright%5Ddx)
represents the red and blue regions combined.
The portion in green is ![\displaystyle \int_{-4}^{8}\left[\sqrt{x+8} - \frac{x}{2} \ \right]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_%7B-4%7D%5E%7B8%7D%5Cleft%5B%5Csqrt%7Bx%2B8%7D%20-%20%5Cfrac%7Bx%7D%7B2%7D%20%5C%20%5Cright%5Ddx)
which is the integral of the difference of the upper and lower curves over the interval 
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In all honesty, it's probably easier to integrate with respect to y since the given functions are in terms of y initially. Also, there isn't a junction point in which the curves swap places in terms of which one is larger. However, it doesn't hurt to have practice in integrating with respect to x.
If you're curious about what the y integral looks like, then it would be
![\displaystyle \int_{-2}^{4} \bigg[ (2y) - (y^2-8)\bigg] dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_%7B-2%7D%5E%7B4%7D%20%5Cbigg%5B%20%282y%29%20-%20%28y%5E2-8%29%5Cbigg%5D%20dy)
You can use a tool like WolframAlpha to check that both integral expressions result in 36 to help confirm that they represent the same overall area (just in different ways of course).
Answer:
The diameter is 6
Step-by-step explanation:
the radius is just half of the diameter so 2x3= 6
Answer:
DOC = 70
BOC = 110
AOD = 110
Step-by-step explanation:
AOB is supplementary to BOC and AOD, which means together they equal 180. 180-70=110, so BOC and AOD are both 110.
AOB and DOC are vertical angles to each other, so are AOD and BOC. Vertical angles are always equal to each other. So since AOB is 70, so is DOC.
Hope this makes sense:)
1.) You have 12 toppings. You choose one topping--that leaves you with 11 toppings. You choose another--that leaves 10. 12×11×10 = 1,320.
Multiply 1,320 topping choices by 6 cheeses to get 7,920 total combinations.
2.) (Though I'm less sure of these)
CDs: 6×5×4×3×2 = 720
Cassettes: 5×4 = 20
DVDs: 8×7×6×5 = 1680
Not really Specific in asking in what way you want it answered... So here is what Im able to understand.
Function
Joseph : 20 + .23x = y
Micki : 10 + .15x =y
Hope this is what your asking...
