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

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

I think i am correct

kjcdb8er | Certified Educator

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