Question

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 5 3 cos(4x) x dx, n

Answer 1

1.1061427

Correct Value (Using calculator) = 0.927007

Explanation:

b=5, a=1, n=8,

h=(b-a)/n=(5-1)/8=0.5

f(x) = 5*cos(4x)/x

The sub-intervals are,

[1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3], [3, 3.5], [3.5, 4], [4, 4.5], [4.5, 5]

Let the integral be denoted by I.

a) From Trapezoidal Rule:

I = [h/2] { f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + 2f(x4) + 2f(x5) + 2(x6) + 2f(x7) + f(x8) }

= [0.5 / 2] { f(1) + 2f(1.5) + 2f(2) + 2f(2.5) + 2f(3) + 2f(3.5) + 2f(4) + 2f(4.5) + f(5) }

= [0.25] { -3.2682 + 2*3.2005 + 2*-0.36375 + 2*-1.6781 + 2*1.4064 + 2*0.19533 + 2*-1.1970 + 2*0.7336 + 0.40808}

= 0.433489

b) From Midpoint Rule:

Values are calculated at midpoint of each interval

I = [h] { f(x1*) + f(x2*) + f(x3*) + f(x4*) + f(x5*) + (x6*) + f(x7*) + f(x8*) }

= [0.5] { f(1.25) + f(1.75) + f(2.25) + f(2.75) + f(3.25) + f(3.75) + f(4.25) + f(4.75) }

= 1.186071

c) Simpson's Rule:

I = [h/3] { f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + 4f(x5) + 2(x6) + 4f(x7) + f(x8) }

= [0.5 / 3] { f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + 2f(3) + 4f(3.5) + 2f(4) + 4f(4.5) + f(5) }

= [0.25] { -3.2682 + 4*3.2005 + 2*-0.36375 + 4*-1.6781 + 2*1.4064 + 4*0.19533 + 2*-1.1970 + 4*0.7336 + 0.40808}

= 1.1061427

Correct Value (Using calculator) = 0.927007