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Use the Midpoint Rule with the given value of n to approximate the integral. Round the...

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tgl223 | Student, College Freshman | (Level 3) eNoter

Posted December 3, 2010 at 8:53 AM via web

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Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.

int_1^5 3 x^3 e^(-2 x) dx text(, ) n = 4
M4 = ?

I got the answer 0.9711, but it keeps saying its wrong. Why?

I think i am correct

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kjcdb8er | Teacher | (Level 1) Associate Educator

Posted December 3, 2010 at 9:21 AM (Answer #1)

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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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