Estimate the integral $$ \int_{1}^{7} \sqrt{x} d x $$ using a left-hand sum and a right-hand sum with $$ n=3 $$ subdivisions. NOTE: Enter exact answers or round to two decimal places. Left-hand sum $$ =\square $$ Right-hand sum $$ = $$

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To estimate the integral $$ \int_{1}^{7} \sqrt{x} \, dx $$ using left-hand and right-hand sums with $$ n = 3 $$ subdivisions, we first need to determine the width of each subdivision.

1. **Calculate the width of each subdivision:**
   $$
   \Delta x = \frac{b - a}{n} = \frac{7 - 1}{3} = \frac{6}{3} = 2
   $$

2. **Determine the endpoints of the subdivisions:**
   The intervals will be:
   - From $$ x_0 = 1 $$ to $$ x_1 = 3 $$
   - From $$ x_1 = 3 $$ to $$ x_2 = 5 $$
   - From $$ x_2 = 5 $$ to $$ x_3 = 7 $$

3. **Left-hand sum:**
   For the left-hand sum, we evaluate the function $$ \sqrt{x} $$ at the left endpoints of each interval:
   - $$ f(x_0) = \sqrt{1} = 1 $$
   - $$ f(x_1) = \sqrt{3} $$
   - $$ f(x_2) = \sqrt{5} $$

The left-hand sum $$ L_n $$ is given by:
   $$
   L_n = \Delta x \left( f(x_0) + f(x_1) + f(x_2) \right) = 2 \left( \sqrt{1} + \sqrt{3} + \sqrt{5} \right)
   $$
   Now we calculate:
   $$
   L_n = 2 \left( 1 + \sqrt{3} + \sqrt{5} \right)
   $$
   Approximating $$ \sqrt{3} \approx 1.732 $$ and $$ \sqrt{5} \approx 2.236 $$:
   $$
   L_n \approx 2 \left( 1 + 1.732 + 2.236 \right) = 2 \left( 5.968 \right) \approx 11.936
   $$

4. **Right-hand sum:**
   For the right-hand sum, we evaluate the function $$ \sqrt{x} $$ at the right endpoints of each interval:
   - $$ f(x_1) = \sqrt{3} $$
   - $$ f(x_2) = \sqrt{5} $$
   - $$ f(x_3) = \sqrt{7} $$

The right-hand sum $$ R_n $$ is given by:
   $$
   R_n = \Delta x \left( f(x_1) + f(x_2) + f(x_3) \right) = 2 \left( \sqrt{3} + \sqrt{5} + \sqrt{7} \right)
   $$
   Approximating $$ \sqrt{7} \approx 2.646 $$:
   $$
   R_n \approx 2 \left( 1.732 + 2.236 + 2.646 \right) = 2 \left( 6.614 \right) \approx 13.228
   $$

5. **Final results:**
   - Left-hand sum $$ L_n \approx 11.94 $$ (rounded to two decimal places)
   - Right-hand sum $$ R_n \approx 13.23 $$ (rounded to two decimal places)

Thus, the final answers are:
$$
\text{Left-hand sum} = 11.94
$$
$$
\text{Right-hand sum} = 13.23
$$
Left-hand sum $$ = 11.94 $$ Right-hand sum $$ = 13.23 $$

Estimate the integral using a left-hand sum and a
right-hand sum with subdivisions.
NOTE: Enter exact answers or round to two decimal places.
Left-hand sum
Right-hand sum

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Estimate the integral using a left-hand sum and a right-hand sum with subdivisions. NOTE: Enter exact answers or round to two decimal places. Left-hand sum Right-hand sum