Trapezoidal Rule Question (Cal 2)?

Question

Use the Trapezoidal Rule with step size n=4 to estimate:

2
∫ (t^3 +t) dx
0


Please show steps. Thank you for the help.

Answer 1

f(t) = t^3 + t
First, find the subinterval length:

[2 - 0] / 4 = .5

Then find the base lengths for each trapezoid:

a ---> f(0) , f(.5) ---> [0 , 0.625]
b ---> f(.5) , f(1) ---> [0.625 , 2]
c ---> f(1) , f(1.5) ---> [2 , 4.875]
d ----> f(1.5) , f(2) ---> [4.875, 10]

The area of a trapezoid is (1/2)(base 1 + base 2)*height

Area of a = (1/2)(0 + 0.625)*.5
Area of b = (1/2)(0.625 + 2)*.5
Area of c = (1/2)(2 + 4.875)*.5
Area of d = (1/2)(4.875 + 10)*.5

Integral {0 to 2} [t^3 + t]dt = Sum of the 4 areas =

=(1/2)^2 * (0 + 2*(0.625) + 2*(2) + 2*(4.875) + 10)
=6.25

Answer 2

2
∫ (t^3 +t) dx ≈ (2-0)/(2*4) [ f(0) + 2*f(0.5) + 2*f(1) + 2*f(1.5) + f(2) ]
0

= (1/4) [ 0 + 1.25 + 4 + 9.75 + 10 ] = 25/4 = 6.25

By integration we can check the exact value of the area:

2
∫ (t^3 +t) dx =
0

2
[ (t^4)/4 + (t^2)/2 ] = 6.0
0

Answer 3

Compute t^3+t at 0, 0.5, 1 , 1.5 and 2. Starting at x=0.5, find 0.25 (t^3+t at x + t*3+t at x-0.5)= Atrap. Do the same for x=1, x=1.5 and 2. Add the "Atraps"