Sunday, September 28, 2014

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. M4 = ? I got the...

The midpoint rule basically estimates the area under a
curve by sampling it a n-1 points, and using the rectangle with height f(Xn) and width
(b-a)/n at each point to give the area under that portion of the
curve.


So,


f(x) = 3x^3 e^-2x ,
 x = [1,5]


dx = 5-1 / 4 =
1


The first point is the midpoint between a and a + dx : a
+ dx/2, or 1.5. Each point after that is spaced at dx =
1.


f(1.5) = 0.68


f(2.5)
= 5.29


f(3.5) = 20.3


f(4.5)
= 55.5


So,


Int( f(x) ) , x =
[1,5] ~~ dx*(0.68 + 5.29 + 20.3 + 55.5) =
81.77



You can compare this with the exact
answer,


3124/(5 e^2) ~~ 84.5575

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