To find an underestimate for the integral [tex]\(\int_2^8 f(x) \, dx\)[/tex] using a Riemann sum with [tex]\(n = 3\)[/tex] subdivisions, we need to follow these steps:
1. Identify the interval and number of subdivisions:
- We are integrating from [tex]\(x = 2\)[/tex] to [tex]\(x = 8\)[/tex].
- The number of subdivisions, [tex]\(n\)[/tex], is given as 3.
2. Determine the width of each sub-interval:
- The interval length is [tex]\(8 - 2 = 6\)[/tex].
- With [tex]\(n = 3\)[/tex] subdivisions, the width of each sub-interval, [tex]\(\Delta x\)[/tex], is given by:
[tex]\[
\Delta x = \frac{8 - 2}{3} = 2.0
\][/tex]
3. List the [tex]\(x\)[/tex]-values at the endpoints of the sub-intervals:
- The sub-intervals will be [tex]\([2, 4]\)[/tex], [tex]\([4, 6]\)[/tex], and [tex]\([6, 8]\)[/tex].
4. Identify the left endpoints and corresponding [tex]\(f(x)\)[/tex] values for the sub-intervals:
- For an underestimate, we use the left endpoints (since [tex]\(f(x)\)[/tex] is decreasing).
- The [tex]\(x\)[/tex]-values at the left endpoints are [tex]\(2\)[/tex], [tex]\(4\)[/tex], and [tex]\(6\)[/tex].
- The corresponding [tex]\(f(x)\)[/tex]-values are [tex]\(f(2) = 64\)[/tex], [tex]\(f(4) = 45\)[/tex], and [tex]\(f(6) = 25\)[/tex].
5. Compute the Riemann sum:
- The Riemann sum is given by:
[tex]\[
\text{Riemann sum} = \Delta x \cdot [f(x_1) + f(x_2) + f(x_3)]
\][/tex]
- Substituting the values we have:
[tex]\[
\text{Riemann sum} = 2.0 \cdot (64 + 45 + 25) = 2.0 \cdot 134 = 268.0
\][/tex]
So, the underestimate for [tex]\(\int_2^8 f(x) \, dx\)[/tex] using a Riemann sum with [tex]\(n=3\)[/tex] subdivisions is:
[tex]\[
\boxed{268}
\][/tex]
