Math / Calculus

QuestionC) Calculate $L_{3}$ and $R_{3}$ for $f(x)=\frac{12}{x}$ on $[2,5]$

Studdy Solution

STEP 1

1. We are given the function $f(x) = \frac{12}{x}$ .
2. We need to calculate the left Riemann sum $L_3$ and the right Riemann sum $R_3$ over the interval $[2, 5]$ .
3. The interval $[2, 5]$ is divided into 3 equal subintervals.

STEP 2

1. Determine the width of each subinterval.
2. Calculate the left Riemann sum $L_3$ .
3. Calculate the right Riemann sum $R_3$ .

STEP 3

Determine the width of each subinterval:
The interval $[2, 5]$ is divided into 3 equal subintervals. The width $\Delta x$ of each subinterval is given by:
$\Delta x = \frac{5 - 2}{3} = 1$

STEP 4

Calculate the left Riemann sum $L_3$ :
The left endpoints of the subintervals are $x_0 = 2$ , $x_1 = 3$ , and $x_2 = 4$ .
\[ L_3 = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) \right] $
Substitute the function values:
$L_3 = 1 \left[ \frac{12}{2} + \frac{12}{3} + \frac{12}{4} \right]$
Simplify:
$L_3 = 1 \left[ 6 + 4 + 3 \right] = 13$

STEP 5

Calculate the right Riemann sum $R_3$ :
The right endpoints of the subintervals are $x_1 = 3$ , $x_2 = 4$ , and $x_3 = 5$ .
\[ R_3 = \Delta x \left[ f(x_1) + f(x_2) + f(x_3) \right] $
Substitute the function values:
$R_3 = 1 \left[ \frac{12}{3} + \frac{12}{4} + \frac{12}{5} \right]$
Simplify:
$R_3 = 1 \left[ 4 + 3 + 2.4 \right] = 9.4$
The values of $L_3$ and $R_3$ are:
$L_3 = 13$ $R_3 = 9.4$

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