Estimate the integral $$ \int_{-1}^{8} 3^{x} d x $$ using a left-hand sum and a right-hand sum with $$ n=3 $$ subdivisions. Left-hand sum $$ =\square $$ Right-hand sum $$ =\square $$

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To estimate the integral $$ \int_{-1}^{8} 3^{x} \, dx $$ using left-hand and right-hand sums with $$ n = 3 $$ subdivisions, we first need to determine the width of each subdivision.

1. **Determine the width of each subdivision:**
   The interval is from $$-1$$ to $$8$$, so the total width is:
   $$
   8 - (-1) = 9
   $$
   With $$ n = 3 $$ subdivisions, the width $$ \Delta x $$ of each subdivision is:
   $$
   \Delta x = \frac{9}{3} = 3
   $$

2. **Determine the endpoints of the subdivisions:**
   The subdivisions will be:
   - From $$-1$$ to $$2$$
   - From $$2$$ to $$5$$
   - From $$5$$ to $$8$$

The left endpoints are:
   - $$ x_0 = -1 $$
   - $$ x_1 = 2 $$
   - $$ x_2 = 5 $$

The right endpoints are:
   - $$ x_1 = 2 $$
   - $$ x_2 = 5 $$
   - $$ x_3 = 8 $$

3. **Calculate the left-hand sum:**
   The left-hand sum $$ L_n $$ is given by:
   $$
   L_n = \Delta x \left( f(x_0) + f(x_1) + f(x_2) \right)
   $$
   where $$ f(x) = 3^x $$.

Evaluating $$ f $$ at the left endpoints:
   - $$ f(-1) = 3^{-1} = \frac{1}{3} $$
   - $$ f(2) = 3^{2} = 9 $$
   - $$ f(5) = 3^{5} = 243 $$

Now, substituting these values into the left-hand sum:
   $$
   L_n = 3 \left( \frac{1}{3} + 9 + 243 \right) = 3 \left( \frac{1}{3} + 9 + 243 \right) = 3 \left( \frac{1}{3} + \frac{27}{3} + \frac{729}{3} \right) = 3 \left( \frac{757}{3} \right) = 757
   $$

4. **Calculate the right-hand sum:**
   The right-hand sum $$ R_n $$ is given by:
   $$
   R_n = \Delta x \left( f(x_1) + f(x_2) + f(x_3) \right)
   $$

Evaluating $$ f $$ at the right endpoints:
   - $$ f(2) = 9 $$
   - $$ f(5) = 243 $$
   - $$ f(8) = 3^{8} = 6561 $$

Now, substituting these values into the right-hand sum:
   $$
   R_n = 3 \left( 9 + 243 + 6561 \right) = 3 \left( 9 + 243 + 6561 \right) = 3 \left( 9 + 243 + 6561 \right) = 3 \left( 6813 \right) = 20439
   $$

5. **Final results:**
   - Left-hand sum $$ = 757 $$
   - Right-hand sum $$ = 20439 $$

Thus, the estimates for the integral are:
$$
\text{Left-hand sum} = 757
$$
$$
\text{Right-hand sum} = 20439
$$
Left-hand sum $$ = 757 $$ Right-hand sum $$ = 20439 $$

Estimate the integral using a left-hand sum and a
right-hand sum with subdivisions.
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. Left-hand sum Right-hand sum