Consider the integral integral from 0 to 1 e superscript 6 x baseline dx with nequals 25 . a. find the trapezoid rule approximations to the integral using n and 2n subintervals. b. find the simpson's rule approximation to the integral using 2n subintervals. c. compute the absolute errors in the trapezoid rule and simpson's rule with 2n subintervals.

a.

With n = 25,

With n = 50,

b.

c.

The absolute error in the trapezoid rule is 0.08047

The absolute error in the Simpson's rule is 0.00008

Step-by-step explanation:

a. To approximate the integral using n = 25 with the trapezoid rule you must:

The trapezoidal rule states that

where

We have that a = 0, b = 1, n = 25.

Therefore,

We need to divide the interval [0,1] into n = 25 sub-intervals of length , with the following endpoints:

Now, we just evaluate the function at these endpoints:

...

Applying the trapezoid rule formula we get

We have that a = 0, b = 1, n = 50.

Therefore,

We need to divide the interval [0,1] into n = 50 sub-intervals of length , with the following endpoints:

Now, we just evaluate the function at these endpoints:

...

Applying the trapezoid rule formula we get

b. To approximate the integral using 2n with the Simpson's rule you must:

The Simpson's rule states that

where

We have that a = 0, b = 1, n = 50

Therefore,

We need to divide the interval [0,1] into n = 50 sub-intervals of length , with the following endpoints:

Now, we just evaluate the function at these endpoints:

...

Applying the Simpson's rule formula we get

c. If B is our estimate of some quantity having an actual value of A, then the absolute error is given by

The absolute error in the trapezoid rule is

The calculated value is

and our estimate is 67.1519320308594

Thus, the absolute error is given by

The absolute error in the Simpson's rule is

step-by-step explanation: