Numerical Integration

I.  Quiz

II.  Homework

III.  Review of Left and Right sums.

 We will review left and right sums in class.  For more information click here

IV.  The Midpoint Approximation

To get a better approximation of , for example, we could instead use the y value of the midpoint of each interval

as the height.  The new numbers are as follows:

2.2 + 0(.5),2.5 + 1(.5), 2.5 + 2(.5), 2.5 + 3(.5), 2.5 + 4(.5), 2.5 + 5(.5) so that the y coordinates are

(2.5 + 0(.5))²,(2.5 + 1(.5))², (2.5 + 2(.5))², (2.5 + 3(.5))², (2.5 + 4(.5))², (2.5 + 5(.5))²

We see that the i^th rectangle has y coordinate:

height = (2.5 + i(.5))² = 6.25 + 2.5i + .25i²

To get the area of the i^th rectangle we multiply the height by the base:

(6.25+ 2.5i + .25i²)(.5)

Finally to get the total area we add the terms up:

sum[(6.25 + 2.5i + .25i²)(.5)]

V.  The Trapezoidal Approximation

A fourth method involves the trapezoidal rule which geometrically calculates the area of the trapezoid with base on the x-axis and heights f(x_i) and f(x_i+1) 

The area of the trapezoid is or the base times the average of the heights.  Adding up all the trapeziods gives

T(n) = f(x₀) + 2f(x₁) + 2f(x₂) +...+ 2f(x_n-1) + f(x_n)

We will demonstrate this in class.

Exercise:  Use the Trapezoidal Approximation to approximate the integral

VI.  Error

The error in approximating an integral can be found by subtracting the true value from the estimated value.  The graphs show that the error is directly inked to the concavity of the integrand.  Without proof, bounds for the errors are:

|E_M| <  B(b - a)³/24n²

|E_T| <  B(b - a)³/12n²

Where B = max |f''(x)|

Example:  If you want to approximate int from 0 to 2 of e^x2 dx using the midpoint rule with an error of less than .001, we compute f''(x) = (2 + 4x²)e^x2 which in an increasing function on [0,2], hence has its maximum at x = 2.  So B = (2 + 4(4))e⁴  < 983

so we find 983(2³)/24n² < .001 or n > 572.  Hence n = 573 will do.

V.  Simpson's Estimate

Yesterday, we saw that the Trapezoidal and Midpoint estimates provided better accuracy than the Left and Right endpoint estimates.  It turns out that a certain combination of the Trapezoid and Midpoint estimates is even better.  

Definition:  Let f(x) be a function defined on [a,b].  Then S(n) = 1/3 T(n) + 2/3 M(n)

where T(n) and M(n) are the Trapezoidal and Midpoint Estimates.

Geometrically, if n is an even number then Simpson's Estimate gives the area under the parabolas defined by connecting three adjacent points.  

Let n be even then using the even subscripted x values for the trapezoidal estimate and the midpoint estimate, gives

S(n) = 1/3[(b - a)/2 (f(x₀) + 2f( f(x₂) + f(x₄)+ ... + f(x_n-2) + f(x_n)]

+ 2/3[f(x₁) + f(x₃) + f(x₅) + ... + f(x_n-1)

= (b - a)/6 ( f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + 4f(x₅) + ... + 2f(x_n-2) + 4f(x_n-1) + f(x_n)

Notice the 1 2 4 2 4 ... 2 4 2 4 1 pattern.

Exercise: Approximate

 VI.  Error in Simpson's Estimate

Without proof, we state

Let M = max |f''''(x)| and let E_S be the error in using Simpson's estimate then

|E_S|  < M(b - a)⁵/180n⁴ 

Examples will be given.