+0  
 
0
46
2
avatar

Let F(x) = -x^2+12x. Let W represent the integral \(\int_0^6 \! F(x) \, \mathrm{d}x.\)

Please evaluate W from the part above as a limit of Riemann Sums.

 

I've been searching for help on this but I cannot find what I am looking for.

 Mar 12, 2021
 #1
avatar+138 
0

Hey there :) I'm not 100% sure what your lesson or teacher is wanting, whether it be a visible graph or just using a calculator, but the limit of the integral from 0 to 6, of F(x), can be modelled by this graph below, If using a calculator, you could simply enter the integral of the equation and let it do the work for you, resulting in an answer of 144 :)

 Mar 12, 2021
 #2
avatar+118069 
+1

Very nice, Space Tsunami   !!!!

 

It appears  that  Space Tsunmai  is  using  the midpoint rule  for  the height of  the rectangles and the width of  a single rectangle  is  .2

 

If we use  this approach we  have  the area  as

 

.2  [  F(.1) + F(.3)  + F(.5) + F(.7) + F(.9) + F(1.1) + F(1.3)  + F(1.5) + F(1.7) + F(1.9)  +F(2.1) +

 

         F(2.3)  +  F (2.5)  +  F(2.7)  + F(2.9)  + F(3.1)  + F(3.3)  + F ( 3.5) + F(3.7) + F(3.9)  + F(4.1) +

 

         F(4.3)  + F(4.5) + F(4.7)  + F(4.9)  + F(5.1) + F(5.3) + F(5.5) + F(5.7) + F(5.9)  ]    =

 

 

.2  [ 1.19  + 3.51 + 5.75 + 7.91  + 9.99  + 11.99  + 13.91   + 15.75  + 17.51  + 19.19  + 20.79 +

 

     22.21  + 23.75  + 25.11  + 26.39 + 27.59  + 28.71 + 29.75  + 30.71  + 31.59  + 32.39 + 33.11 +

 

    33.75  + 34.31  + 34.79 + 35.19 + 35.51  + 35.75 + 35.91  + 35.99  ]       =

 

 

.2  [ 127.49  + 311.31  + 281.2  ]   ≈

 

144

 

 

cool cool cool

CPhill  Mar 12, 2021

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