Integral Aproximation

Question

USe the trapezoidal rule, the midpoints rule, and the simpsons rule to approximate the internal

∫ with upper limit as 13 and lower as 1 (ln(x))/(3+x) DX with n = 4

T₄=

M₄=

S₄=

Answer 1

For the trapezoidal rule, create 4 trapezoids. Divide the space you have in the x-direction equally and since you have 13 - 1 = 12, divide 12 by 4 to get the width for each trapezoid (each will have a width of 3). The height of each side of the trapezoid is the function value. Like this:

Now calculate the area of each trapezoid and add them all up:

A_total = A₁ + A₂ + A₃ + A₄

Each area is the width multiplied by the average of the height (h₁ + h₂)/2. The height is the function value and the width is 3:

A₁ = (3)(f(1) + f(4))/2 = (3)(0 + 0.198)/2 = 0.297

Repeat to get A2, A3, and A4 and then add the results to get the estimated area using the trapezoidal method.

To get the area using the midpoint method, use 4 rectangles, each with a width of 3 and the height will be the function value of the midpoint. The midpoint for the first rectangle is 2.5 (halfway between x = 1 and x = 4).

A_total = A₁ + A₂ + A₃ + A₄

A₁ = (3)(f(2,5)) = 0.1666

Then repeat for each rectangle. A2 = (3)(f(5.5)) = (3)(0.2006) = 0.6017 etc.

To get area using Simpson's Rule, use the following equation:

A = Δx/3[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + . . . f(x_n))]

So, since Δx in this is 3, we get:

A = 3/3[f(1) + 4f(4) + 2f(7) + 4f(10) + f(13)]

A = f(1) + 4f(4) + 2f(7) + 4f(10) + f(13)