To solve this problem, we follow these steps:
### Part (a):
We need to estimate the integral [tex]\(\int_3^9 f(x) \, dx\)[/tex] using three equal subintervals and right endpoints.
#### Step-by-Step Estimation Using Right Endpoints:
1. Determine the interval width:
The interval from [tex]\(3\)[/tex] to [tex]\(9\)[/tex] is divided into three equal parts.
[tex]\[
\Delta x = \frac{9 - 3}{3} = 2
\][/tex]
2. Identify the right endpoints:
For three equal subintervals, the right endpoints are:
[tex]\[
\text{Right endpoints: } 5, 7, 9
\][/tex]
3. Evaluate the function at the right endpoints:
Using the provided table:
[tex]\(\ f(5) = -0.7\)[/tex]
[tex]\(\ f(7) = 0.9\)[/tex]
[tex]\(\ f(9) = 1.7\)[/tex]
4. Calculate the Riemann sum using the right endpoints:
[tex]\[
R_3 = \Delta x \left[ f(5) + f(7) + f(9) \right]
\][/tex]
[tex]\[
R_3 = 2 \left[ (-0.7) + 0.9 + 1.7 \right] = 2 \times 1.9 = 3.8
\][/tex]
So, the estimate using right endpoints is:
[tex]\[
R_3 = 3.8
\][/tex]
5. Determine whether the estimate is less than or greater than the exact value:
Since [tex]\(f(x)\)[/tex] is an increasing function, the right endpoint estimation will be greater than the exact integral.
Hence, the answer is "greater than."
### Part (b):
We need to estimate the integral [tex]\(\int_3^9 f(x) \, dx\)[/tex] using three equal subintervals and left endpoints.
#### Step-by-Step Estimation Using Left Endpoints:
1. Determine the interval width:
The interval from [tex]\(3\)[/tex] to [tex]\(9\)[/tex] is divided into three equal parts.
[tex]\[
\Delta x = \frac{9 - 3}{3} = 2
\][/tex]
2. Identify the left endpoints:
For three equal subintervals, the left endpoints are:
[tex]\[
\text{Left endpoints: } 3, 5, 7
\][/tex]
3. Evaluate the function at the left endpoints:
Using the provided table:
[tex]\(\ f(3) = -3.6\)[/tex]
[tex]\(\ f(5) = -0.7\)[/tex]
[tex]\(\ f(7) = 0.9\)[/tex]
4. Calculate the Riemann sum using the left endpoints:
[tex]\[
L_3 = \Delta x \left[ f(3) + f(5) + f(7) \right]
\][/tex]
[tex]\[
L_3 = 2 \left[ (-3.6) + (-0.7) + 0.9 \right] = 2 \times (-3.4) = -6.8
\][/tex]
So, the estimate using left endpoints is:
[tex]\[
L_3 = -6.8
\][/tex]
5. Determine whether the estimate is less than or greater than the exact value:
Since [tex]\(f(x)\)[/tex] is an increasing function, the left endpoint estimation will be less than the exact integral.
Hence, the answer is "less than."
In summary, the completed responses for the integral estimates and their comparisons are as follows:
(a) Right endpoint estimate:
[tex]\[
R_3 = 3.8 \quad \text{(greater than)}
\][/tex]
(b) Left endpoint estimate:
[tex]\[
L_3 = -6.8 \quad \text{(less than)}
\][/tex]
