Answer:
.
Step-by-step explanation:
To find
.
First, calculate the corresponding indefinite integral:
Integrate term by term:
![\int{\left(- \frac{3 u^{9}}{2} + \frac{2 u^{4}}{5} + 2\right)d u}} =\int{2 d u} + \int{\frac{2 u^{4}}{5} d u} - \int{\frac{3 u^{9}}{2} d u}](https://tex.z-dn.net/?f=%5Cint%7B%5Cleft%28-%20%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20%2B%20%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20%2B%202%5Cright%29d%20u%7D%7D%20%3D%5Cint%7B2%20d%20u%7D%20%2B%20%5Cint%7B%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20d%20u%7D%20-%20%5Cint%7B%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20d%20u%7D)
Apply the constant rule ![\int c\, du = c u](https://tex.z-dn.net/?f=%5Cint%20c%5C%2C%20du%20%3D%20c%20u)
![\int{2 d u}} + \int{\frac{2 u^{4}}{5} d u} - \int{\frac{3 u^{9}}{2} d u} = {\left(2 u\right)} + \int{\frac{2 u^{4}}{5} d u} - \int{\frac{3 u^{9}}{2} d u}](https://tex.z-dn.net/?f=%5Cint%7B2%20d%20u%7D%7D%20%2B%20%5Cint%7B%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20d%20u%7D%20-%20%5Cint%7B%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20d%20u%7D%20%3D%20%7B%5Cleft%282%20u%5Cright%29%7D%20%2B%20%5Cint%7B%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20d%20u%7D%20-%20%5Cint%7B%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20d%20u%7D)
Apply the constant multiple rule ![\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du](https://tex.z-dn.net/?f=%5Cint%20c%20f%7B%5Cleft%28u%20%5Cright%29%7D%5C%2C%20du%20%3D%20c%20%5Cint%20f%7B%5Cleft%28u%20%5Cright%29%7D%5C%2C%20du)
![2 u - {\int{\frac{3 u^{9}}{2} d u}} + \int{\frac{2 u^{4}}{5} d u} = 2 u - {\left(\frac{3}{2} \int{u^{9} d u}\right)} + \left(\frac{2}{5} \int{u^{4} d u}\right)](https://tex.z-dn.net/?f=2%20u%20-%20%7B%5Cint%7B%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20d%20u%7D%7D%20%2B%20%5Cint%7B%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20d%20u%7D%20%3D%202%20u%20-%20%7B%5Cleft%28%5Cfrac%7B3%7D%7B2%7D%20%5Cint%7Bu%5E%7B9%7D%20d%20u%7D%5Cright%29%7D%20%2B%20%5Cleft%28%5Cfrac%7B2%7D%7B5%7D%20%5Cint%7Bu%5E%7B4%7D%20d%20u%7D%5Cright%29)
Apply the power rule ![\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}](https://tex.z-dn.net/?f=%5Cint%20u%5E%7Bn%7D%5C%2C%20du%20%3D%20%5Cfrac%7Bu%5E%7Bn%20%2B%201%7D%7D%7Bn%20%2B%201%7D)
![2 u - \frac{3}{2} {\int{u^{9} d u}} + \frac{2}{5} {\int{u^{4} d u}}=2 u - \frac{3}{2} {\frac{u^{1 + 9}}{1 + 9}}+ \frac{2}{5}{\frac{u^{1 + 4}}{1 + 4}}](https://tex.z-dn.net/?f=2%20u%20-%20%5Cfrac%7B3%7D%7B2%7D%20%7B%5Cint%7Bu%5E%7B9%7D%20d%20u%7D%7D%20%2B%20%5Cfrac%7B2%7D%7B5%7D%20%7B%5Cint%7Bu%5E%7B4%7D%20d%20u%7D%7D%3D2%20u%20-%20%5Cfrac%7B3%7D%7B2%7D%20%7B%5Cfrac%7Bu%5E%7B1%20%2B%209%7D%7D%7B1%20%2B%209%7D%7D%2B%20%5Cfrac%7B2%7D%7B5%7D%7B%5Cfrac%7Bu%5E%7B1%20%2B%204%7D%7D%7B1%20%2B%204%7D%7D)
Therefore,
![\int{\left(- \frac{3 u^{9}}{2} + \frac{2 u^{4}}{5} + 2\right)d u} = - \frac{3 u^{10}}{20} + \frac{2 u^{5}}{25} + 2 u = \frac{u}{100} \left(- 15 u^{9} + 8 u^{4} + 200\right)](https://tex.z-dn.net/?f=%5Cint%7B%5Cleft%28-%20%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20%2B%20%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20%2B%202%5Cright%29d%20u%7D%20%3D%20-%20%5Cfrac%7B3%20u%5E%7B10%7D%7D%7B20%7D%20%2B%20%5Cfrac%7B2%20u%5E%7B5%7D%7D%7B25%7D%20%2B%202%20u%20%3D%20%5Cfrac%7Bu%7D%7B100%7D%20%5Cleft%28-%2015%20u%5E%7B9%7D%20%2B%208%20u%5E%7B4%7D%20%2B%20200%5Cright%29)
According to the Fundamental Theorem of Calculus,
, so just evaluate the integral at the endpoints, and that's the answer.
![\left(\frac{u}{100} \left(- 15 u^{9} + 8 u^{4} + 200\right)\right)|_{\left(u=1\right)}=\frac{193}{100}](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7Bu%7D%7B100%7D%20%5Cleft%28-%2015%20u%5E%7B9%7D%20%2B%208%20u%5E%7B4%7D%20%2B%20200%5Cright%29%5Cright%29%7C_%7B%5Cleft%28u%3D1%5Cright%29%7D%3D%5Cfrac%7B193%7D%7B100%7D)
![\left(\frac{u}{100} \left(- 15 u^{9} + 8 u^{4} + 200\right)\right)|_{\left(u=0\right)}=0](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7Bu%7D%7B100%7D%20%5Cleft%28-%2015%20u%5E%7B9%7D%20%2B%208%20u%5E%7B4%7D%20%2B%20200%5Cright%29%5Cright%29%7C_%7B%5Cleft%28u%3D0%5Cright%29%7D%3D0)
![\int_{0}^{1}\left( - \frac{3 u^{9}}{2} + \frac{2 u^{4}}{5} + 2 \right)du=\left(\frac{u}{100} \left(- 15 u^{9} + 8 u^{4} + 200\right)\right)|_{\left(u=1\right)}-\left(\frac{u}{100} \left(- 15 u^{9} + 8 u^{4} + 200\right)\right)|_{\left(u=0\right)}=\frac{193}{100}](https://tex.z-dn.net/?f=%5Cint_%7B0%7D%5E%7B1%7D%5Cleft%28%20-%20%5Cfrac%7B3%20u%5E%7B9%7D%7D%7B2%7D%20%2B%20%5Cfrac%7B2%20u%5E%7B4%7D%7D%7B5%7D%20%2B%202%20%5Cright%29du%3D%5Cleft%28%5Cfrac%7Bu%7D%7B100%7D%20%5Cleft%28-%2015%20u%5E%7B9%7D%20%2B%208%20u%5E%7B4%7D%20%2B%20200%5Cright%29%5Cright%29%7C_%7B%5Cleft%28u%3D1%5Cright%29%7D-%5Cleft%28%5Cfrac%7Bu%7D%7B100%7D%20%5Cleft%28-%2015%20u%5E%7B9%7D%20%2B%208%20u%5E%7B4%7D%20%2B%20200%5Cright%29%5Cright%29%7C_%7B%5Cleft%28u%3D0%5Cright%29%7D%3D%5Cfrac%7B193%7D%7B100%7D)
M=square root of 10 all over 2
Perpendicular means "at right angles". A line meeting another at a right angle, or 90° is said to be perpendicular to it. So just to find the line segment that fits that description. Hope this helps!