Answer:
a)550
b)910
c)730
Step-by-step explanation:
The given model is
![v(t) = 10t^2 ft/s](https://tex.z-dn.net/?f=v%28t%29%20%3D%2010t%5E2%20ft%2Fs)
Use the interval [0,6], with n=6 rectangles
Then, the interval width is
![\Delta t = \frac{b-a}{n}](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D)
= 1
so, the sub intervals are
[0,1], [1,2], [2,3], [3,4],[4,5],[5,6]
Now evaluating the function values
![f(t_0)= f(0) = 0](https://tex.z-dn.net/?f=f%28t_0%29%3D%20f%280%29%20%3D%200)
![f(t_1)= f(1) = 10](https://tex.z-dn.net/?f=f%28t_1%29%3D%20f%281%29%20%3D%2010)
![f(t_2)= f(2) = 40](https://tex.z-dn.net/?f=f%28t_2%29%3D%20f%282%29%20%3D%2040)
![f(t_3)= f(3) = 90](https://tex.z-dn.net/?f=f%28t_3%29%3D%20f%283%29%20%3D%2090)
![f(t_4)= f(4) = 160](https://tex.z-dn.net/?f=f%28t_4%29%3D%20f%284%29%20%3D%20160)
![f(t_5)= f(5) = 250](https://tex.z-dn.net/?f=f%28t_5%29%3D%20f%285%29%20%3D%20250)
![f(t_6)= f(6) = 360](https://tex.z-dn.net/?f=f%28t_6%29%3D%20f%286%29%20%3D%20360)
a) left hand sum is
L_6 = ![\Delta t [f(t_0)+ f(t_1)+f(t_2)+f(t_3)+f(t_4)+f(t_5)]](https://tex.z-dn.net/?f=%5CDelta%20t%20%5Bf%28t_0%29%2B%20f%28t_1%29%2Bf%28t_2%29%2Bf%28t_3%29%2Bf%28t_4%29%2Bf%28t_5%29%5D)
=![1 [0+ 10+40+90+160+250]](https://tex.z-dn.net/?f=1%20%5B0%2B%2010%2B40%2B90%2B160%2B250%5D)
= 550
b) right hand sum
R_6 = ![\Delta t [ f(t_1)+f(t_2)+f(t_3)+f(t_4)+f(t_5)+f(t_6)]](https://tex.z-dn.net/?f=%5CDelta%20t%20%5B%20f%28t_1%29%2Bf%28t_2%29%2Bf%28t_3%29%2Bf%28t_4%29%2Bf%28t_5%29%2Bf%28t_6%29%5D)
= ![1 [10+40+90+160+250+360]](https://tex.z-dn.net/?f=1%20%5B10%2B40%2B90%2B160%2B250%2B360%5D)
= 910
c) average of two sums is
![\frac{L_5+R_5}{2}](https://tex.z-dn.net/?f=%5Cfrac%7BL_5%2BR_5%7D%7B2%7D)
= ![\frac{550+910}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B550%2B910%7D%7B2%7D)
=730