Consider the function, f(x) = 1 1 + x 2 .
a. (10 pts) compute z 1 −1 f(x) dx using five point simpson’s method (two simpson partitions). show all the calculations in exact arithmetic (i. e. use fractions throughout).
b. (15 pts) partition the interval [−1, 1] into 5 subintervals. for this partition, compute z 1 −1 f(x) dx using trapezoid and midpoint methods. show all the calculations in exact arithmetic

a. Simpson method

b. Trapezoid method

Midpoint method

Step-by-step explanation:

a. We want to compute using five point Simpson’s method.

The Simpson's rule states that

where

We know that a = −1, b = 1, n = 4.

Therefore, .

Divide the interval [−1,1] into n = 4 sub-intervals of length

Now, we just evaluate the function at these endpoints:

Finally, just sum up the above values and multiply by

.

b.

The trapezoidal rule states that

where

We know that a = −1, b = 1, n = 5.

Therefore, .

Divide the interval [−1,1] into n = 5 sub-intervals of length

Now, we just evaluate the function at these endpoints:

The Midpoint method uses the midpoints of sub-interval:

where

We know that a = −1, b = 1, n = 5.

Therefore, .

Divide the interval [−1,1] into n = 5 sub-intervals of length

Now, we just evaluate the function at these endpoints: