Answer:
110
Step-by-step explanation:
Each cross section has side length equal to
satisfying
, where
so that
.
The exact volume is given by the definite integral,
![\displaystyle\int_1^4(\ln y)^2\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_1%5E4%28%5Cln%20y%29%5E2%5C%2C%5Cmathrm%20dy)
Take a slice at any value of
with thickness
. Then the slice has volume
.
The approximate total volume of these slices is then given by the Riemann sum,
![\displaystyle\sum_{i=1}^n(\ln y_i)^2\Delta y_i](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bi%3D1%7D%5En%28%5Cln%20y_i%29%5E2%5CDelta%20y_i)
where
are chosen however you like from the range above.
Compute the definite integral above for the exact volume: you can do this by parts, taking
![u=(\ln y)^2\implies\mathrm du=\dfrac{2\ln y}y\,\mathrm dy](https://tex.z-dn.net/?f=u%3D%28%5Cln%20y%29%5E2%5Cimplies%5Cmathrm%20du%3D%5Cdfrac%7B2%5Cln%20y%7Dy%5C%2C%5Cmathrm%20dy)
![\mathrm dv=\mathrm dy\implies v=y](https://tex.z-dn.net/?f=%5Cmathrm%20dv%3D%5Cmathrm%20dy%5Cimplies%20v%3Dy)
![\implies\displaystyle\int_1^4(\ln y)^2\,\mathrm dy=y(\ln y)^2\bigg|_1^4-2\int_1^4\ln y\,\mathrm dy](https://tex.z-dn.net/?f=%5Cimplies%5Cdisplaystyle%5Cint_1%5E4%28%5Cln%20y%29%5E2%5C%2C%5Cmathrm%20dy%3Dy%28%5Cln%20y%29%5E2%5Cbigg%7C_1%5E4-2%5Cint_1%5E4%5Cln%20y%5C%2C%5Cmathrm%20dy)
The remaining integral can be done by parts again, this time with
![u=\ln y\implies\mathrm du=\dfrac{\mathrm dy}y](https://tex.z-dn.net/?f=u%3D%5Cln%20y%5Cimplies%5Cmathrm%20du%3D%5Cdfrac%7B%5Cmathrm%20dy%7Dy)
![\mathrm dv=\mathrm dy\implies v=y](https://tex.z-dn.net/?f=%5Cmathrm%20dv%3D%5Cmathrm%20dy%5Cimplies%20v%3Dy)
![\implies\displaystyle\int_1^4\ln y\,\mathrm dy=y\ln y\bigg|_1^4-\int_1^4\mathrm dy](https://tex.z-dn.net/?f=%5Cimplies%5Cdisplaystyle%5Cint_1%5E4%5Cln%20y%5C%2C%5Cmathrm%20dy%3Dy%5Cln%20y%5Cbigg%7C_1%5E4-%5Cint_1%5E4%5Cmathrm%20dy)
and of course
![\displaystyle\int_1^4\mathrm dy=y\bigg|_1^4](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_1%5E4%5Cmathrm%20dy%3Dy%5Cbigg%7C_1%5E4)
So we have
![\displaystyle\int_1^4(\ln y)^2\,\mathrm dy=4(\ln 4)^2-2(4\ln 4-(4-1))](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_1%5E4%28%5Cln%20y%29%5E2%5C%2C%5Cmathrm%20dy%3D4%28%5Cln%204%29%5E2-2%284%5Cln%204-%284-1%29%29)
![\displaystyle\int_1^4(\ln y)^2\,\mathrm dy=4(\ln 4)^2-8\ln 4+6](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_1%5E4%28%5Cln%20y%29%5E2%5C%2C%5Cmathrm%20dy%3D4%28%5Cln%204%29%5E2-8%5Cln%204%2B6)
Answer:
(x+2)/(x+3)
Step-by-step explanation:
because same denominator(x+2) and numerator(x+2) can be removed.
Answer:
y=-4x+7/2
Step-by-step explanation:
slope-intercept form follows the following format:
y=mx+b
simply replace m with -4 and b with 7/2