Chapter 6.5, Problem 22E

To determine

Graphically represent a function shows Trapezoidal Rule obviously gives a better approximation than Simpson’s Rule for $n=4$ .

Explanation of Solution

Given information:

For a given value of n,

Simpson’s Rule gives a better approximation to an integral than the Trapezoidal Rule.

Since the Simpson’s Rule approximates the curve using parabolic arches which fit the curves of the function better than trapezoids.

Thus,

Simpson’s Rule gives a better approximation.

However,

If a function is made up of line segments instead of curves, the Trapezoidal rule will give a better approximation.

The graph below shows a possible example:

Divide the graph into four subintervals would give four trapezoids.

Thus,

The Trapezoidal Rule would give an exact value for the area while Simpson’s Rule would give an estimate.