Answer:
13
Step-by-step explanation:
8 - (-5) = 13
Since x is the same in both points we only have to count the y
![y=\displaystyle\int_1^x\sqrt{t^3-1}\,\mathrm dt](https://tex.z-dn.net/?f=y%3D%5Cdisplaystyle%5Cint_1%5Ex%5Csqrt%7Bt%5E3-1%7D%5C%2C%5Cmathrm%20dt)
By the fundamental theorem of calculus,
![\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm d}{\mathrm dx}\displaystyle\int_1^x\sqrt{t^3-1}\,\mathrm dt=\sqrt{x^3-1}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cdisplaystyle%5Cint_1%5Ex%5Csqrt%7Bt%5E3-1%7D%5C%2C%5Cmathrm%20dt%3D%5Csqrt%7Bx%5E3-1%7D)
Now the arc length over an arbitrary interval
![(a,b)](https://tex.z-dn.net/?f=%28a%2Cb%29)
is
![\displaystyle\int_a^b\sqrt{1+\left(\frac{\mathrm dy}{\mathrm dx}\right)^2}\,\mathrm dx=\int_a^b\sqrt{1+x^3-1}\,\mathrm dx=\int_a^bx^{3/2}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_a%5Eb%5Csqrt%7B1%2B%5Cleft%28%5Cfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cright%29%5E2%7D%5C%2C%5Cmathrm%20dx%3D%5Cint_a%5Eb%5Csqrt%7B1%2Bx%5E3-1%7D%5C%2C%5Cmathrm%20dx%3D%5Cint_a%5Ebx%5E%7B3%2F2%7D%5C%2C%5Cmathrm%20dx)
But before we compute the integral, first we need to make sure the integrand exists over it.
![x^{3/2}](https://tex.z-dn.net/?f=x%5E%7B3%2F2%7D)
is undefined if
![x](https://tex.z-dn.net/?f=x%3C0)
, so we assume
![a\ge0](https://tex.z-dn.net/?f=a%5Cge0)
and for convenience that
![a](https://tex.z-dn.net/?f=a%3Cb)
. Then
The answer to c is 9$ for four "cartons" of soda the answer to a is 1560 mL
Answer:
Mean = 30.68
Standard deviation = 6.095
Two standard deviations above the mean = 42.87
Step-by-step explanation:
Given the following data :
21; 21; 22; 23; 24; 24; 25; 25; 28; 29; 29; 31; 32; 33; 33; 34; 35; 36; 36; 36; 36; 38; 38; 38; 40
Mean = Σ(x) / n
n = 25
Σ (X) = 767
Mean (m) = 767 / 25
Mean (m) = 30.68
Standard deviation : sqrt[Σ(x - m) / n]
Using a calculator :
Standard deviation = 6.095 ( 2 decimal places)
2 standard deviations above the mean :
Mean + 2(standard deviation)
Mean + 2(6.095)
30.68 + 12.19
30.68 + 12.19
= 42.87