Approximating Definite Integrals — Question 8

Question 8

For an increasing function on $[0,3]$, suppose $$L_6=7.2,\qquad R_6=8.7.$$ Find $T_6$ and bound the integral.

Question 8 – Solution

Step 1: Use the trapezoidal-average identity. With equal-width subintervals, $$T_n=\frac{L_n+R_n}{2}.$$ Thus $$\begin{align*} T_6&=\frac{7.2+8.7}{2}\\ &=\frac{15.9}{2}=7.95. \end{align*}$$ Step 2: Use monotonicity to bound the integral. Because $f$ is increasing, left endpoints underestimate and right endpoints overestimate: $$L_6\le I\le R_6.$$ Therefore, $$7.2\le I\le8.7.$$ $$\boxed{7.2\le I\le8.7,\quad T_6=7.95}$$