Chapter 6.5, Problem 36E

To determine

Correct order of approximations using the same number of equal subdivisions of [ a , b ].

Answer to Problem 36E

Correct order:

  $L<I<T<R$

Explanation of Solution

Given information:

On interval [ a , b ],

  $f,f\text{'},f"\text{are}\text{all}\text{positive}$ .

  $I={\displaystyle {\int }_a^{b}f\left(x\right)dx}$

Such that

L : LRAM approximation

R : RRAM approximation

T : Trapezoidal approximation

We know that

For [ a , b ],

  $f>0$

And

  $f\text{'}>0$

Thus,

f is strictly increasing.

When a positive function is strictly increasing,

The right Riemann sum is an overestimate.

Such that

  $I<R$

The left Riemann sum is an underestimate.

Such that

  $L<I$

Thus,

  $L<I<R$

Now,

We also know that

For [ a , b ],

  $f">0$

Thus,

f is concave up.

When a positive function is concave up,

The trapezoidal rule gives an overestimate.

Thus,

  $I<T$

Since the trapezoidal rule is the average of left Riemann sum and right Riemann sum,

T must lie between R and L .

Therefore,

The correct order is:

  $L<I<T<R$