Problem 58AE

Shortcut for the Trapezoid Rule

Prove that if you have M(n)and T(n)(a Midpoint Rule approximation and a Trapezoid Rule approximation with n subintervals), then T(2n) = (T(n) + M(n))/2.

Solution:-

Step1

Given that

A Midpoint Rule approximation and a Trapezoid Rule approximation with n subintervals.

Step2

Prove that

Step3

The Midpoint Rule is the approximation of

+-------+]

The Trapezoidal Rule is the approximation of

2+-------+]

We can divide the interval [a,b] into 2n equally spaced , we get

+-------+]

And

So, using the midpoint formula this become

2+-------+]

Similarly, The Trapezoidal Rule may be written as

2+-------+]

Multiply both by and add we get

2+-------+]

And using the definition of Trapezoidal Rule we find

=

Hence,

Proved