Chapter 5.5, Problem 36E

To determine

To choose : The correct option.

Answer to Problem 36E

The answer is: $(C)$

Explanation of Solution

Given information:

  $f,f\text{'},f\text{'}\text{'}$ are all positive on the interval $\left[a,b\right]$ .

The Trapezoidal approximation of $I={{\displaystyle \int }}_a^{2}f(x)dx$ to use the same number of equal subdivision of $\left[a,b\right]$ .

Calculation:

Keep in mind that LRAM and RRAM are Reimann approximations that utilise, respectively, the left and right end points.

  $(1)$ The right and left end points are the highest and lowest positions in each interval since the function is rising.

Therefore, $L<I<R$

  $(2)$ Trapezoidal sum $=\frac{{\text{ LRAM }}_n+{\text{ RRAM }}_n}{2}$

Therefore, $T$ is between $L$ and $R$ .

  $(3)$ The graph is concave up and the midpoint of each interval is above the curve since the second derivative is positive.

Therefore, $I<T$

To use $(1),(2)$ and $(3)$ , write that,

  $L<I<T<R$

Therefore, the required correct option is $(C)$ .