Chapter 5, Problem 1RE

(a)

To determine

The Riemann sum with six subintervals from the graph at the left endpoints.

Answer to Problem 1RE

The Riemann sum with six subintervals at the left endpoints is 8.

Explanation of Solution

Given information:

Take the curve as $y=f\left(x\right)$.

The region lies between $x=0$ and $x=6$. Hence, the limits are $a=0$ and $b=6$.

Consider that the Number of rectangles as $n=6$.

The expression to find the Riemann sum at the left endpoint $\left(L_n\right)$ is shown below:

$L_n={\displaystyle \sum _{i=1}^{n}f\left(x_{i-1}\right)\Delta x}$ (1)

Here, the height of sample at the left endpoint is $f\left(x_{i-1}\right)$ and the width is $\Delta x$.

Find the width $\left(\Delta x\right)$ using the relation:

$\Delta x=\frac{b-a}{n}$ (2)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 6 for b, 0 for a and 6 for n in Equation (2).

$\begin{array}{c}\Delta x=\frac{6-0}{6}\\ =1\end{array}$

Draw six rectangles at the left endpoints as shown in Figure (1).

Calculate $L_6$ using Equation (1).

Substitute 6 for n in equation (1).

$L_6={\displaystyle \sum _{i=1}^{6}f\left(x_{i-1}\right)\Delta x}$ (3)

Substitute 1, 2, 3, 4, 5, and 6 for i in equation (3).

$L_6=f\left(x_0\right)\Delta x+f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+f\left(x_3\right)\Delta x+f\left(x_4\right)\Delta x+f\left(x_5\right)\Delta x$ (4)

Refer to Figure (1).

Take the height of the sample point values of $f\left(x_0\right)$ as 2, $f\left(x_1\right)$ as 3.5, $f\left(x_2\right)$ as 4, $f\left(x_3\right)$ as 2, $f\left(x_4\right)$ as $-1$ and $f\left(x_5\right)$ as $-2.5$.

Substitute 1 for $\Delta x$, 2 for $f\left(x_0\right)$, 3.5 for $f\left(x_1\right)$, 4 for $f\left(x_2\right)$, 2 for $f\left(x_3\right)$, $-1$ for $f\left(x_4\right)$ and $-2.5$ for $f\left(x_5\right)$ in Equation (4).

$\begin{array}{c}{L}_6=2\times 1+3.5\times 1+4\times 1+2\times 1+\left(-1\right)\times 1+\left(-2.5\right)\times 1\\ =8\end{array}$

Therefore, the Riemann sum with six subintervals at the left endpoint is 8.

(b)

To determine

The Riemann sum with six subintervals from the graph at the midpoints.

Answer to Problem 1RE

The Riemann sum with six subintervals at the midpoints are 5.7.

Explanation of Solution

Given information:

Take the curve as $y=f\left(x\right)$.

The region lies between $x=0$ and $x=6$. Hence, the limits are $a=0$ and $b=6$.

Consider the number of rectangles to be $n=6$.

The expression to find the Riemann sum at the midpoint $\left(M_n\right)$ is shown below:

$M_n={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)\Delta x}$ (5)

Here, the height of the sample point at the midpoint is $f\left({\overline{x}}_i\right)$ and the width is $\Delta x$.

Substitute 6 for b, 0 for a, and 6 for n in Equation (2).

$\begin{array}{c}\Delta x=\frac{b-a}{n}\\ =\frac{6-0}{6}\\ =1\end{array}$

Draw six rectangles at the midpoints as shown in Figure (2).

Calculate $M_6$ using Equation (5).

Substitute 6 for n in equation (5).

$M_6={\displaystyle \sum _{i=1}^{6}f\left({\overline{x}}_i\right)\Delta x}$ (6)

Substitute 1, 2, 3, 4, 5, and 6 for i in equation (6).

$M_6=f\left({\overline{x}}_1\right)\Delta x+f\left({\overline{x}}_2\right)\Delta x+f\left({\overline{x}}_3\right)\Delta x+f\left({\overline{x}}_4\right)\Delta x+f\left({\overline{x}}_5\right)\Delta x+f\left({\overline{x}}_6\right)\Delta x$ (7)

Substitute 0.5 for ${\overline{x}}_1$, 1.5 for ${\overline{x}}_2$, 2.5 for ${\overline{x}}_3$, 3.5 for ${\overline{x}}_4$, 4.5 for ${\overline{x}}_5,$ and 5.5 for ${\overline{x}}_6$ in equation (7).

$M_6=f\left(0.5\right)\Delta x+f\left(1.5\right)\Delta x+f\left(2.5\right)\Delta x+f\left(3.5\right)\Delta x+f\left(4.5\right)\Delta x+f\left(5.5\right)\Delta x$ (8)

Refer to Figure (2).

Take the height of the sample point at midpoint values of $f\left(0.5\right)$ as 3, $f\left(1.5\right)$ as 3.9, $f\left(2.5\right)$ as 3.4, $f\left(3.5\right)$ as 0.3, $f\left(4.5\right)$ as $-2$. and $f\left(5.5\right)$ as $-2.9$.

Substitute 1 for $\Delta x$, 3 for $f\left(0.5\right)$, 3.9 for $f\left(1.5\right)$, 3.4 for $f\left(2.5\right)$, 0.3 for $f\left(3.5\right)$, $-2$ for $f\left(4.5\right)$ and $-2.9$ for $f\left(5.5\right)$ in Equation (8).

$\begin{array}{c}{M}_6=3\times 1+3.9\times 1+3.4\times 1+0.3\times 1+\left(-2\right)\times 1+\left(-2.9\right)\times 1\\ =5.7\end{array}$

Therefore, the Riemann sum with six subintervals at the midpoint is 5.7.