Question

question let f be the function graphed below. if three subintervals of equal length are used, draw rectangles whose area represents a right riemann sum approximation of \\(\int_{1}^{8.5} f(x) dx\\). width of each subinterval:

Explanation:

Step1: Calculate subinterval width

The formula for the width of each subinterval is $\Delta x = \frac{b-a}{n}$, where $a=1$, $b=8.5$, $n=3$.
$\Delta x = \frac{8.5 - 1}{3} = \frac{7.5}{3} = 2.5$

Step2: Identify subinterval endpoints

The subintervals are $[1, 3.5]$, $[3.5, 6]$, $[6, 8.5]$. For right Riemann sum, use right endpoints $x_1=3.5$, $x_2=6$, $x_3=8.5$.
(Note: To draw the rectangles:

1. For $[1, 3.5]$, draw a rectangle with width 2.5, height $f(3.5)$ (from the graph), left at $x=1$, right at $x=3.5$, top at $y=f(3.5)$.
2. For $[3.5, 6]$, draw a rectangle with width 2.5, height $f(6)$, left at $x=3.5$, right at $x=6$, top at $y=f(6)$.
3. For $[6, 8.5]$, draw a rectangle with width 2.5, height $f(8.5)$, left at $x=6$, right at $x=8.5$, top at $y=f(8.5)$.)

Answer:

Width of each subinterval: $2.5$
(To complete the right Riemann sum rectangles: draw three rectangles over subintervals $[1,3.5]$, $[3.5,6]$, $[6,8.5]$ with heights equal to the function's value at the right endpoint of each subinterval.)