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 \( =\square \)

To estimate the integral \( \int_{-1}^{8} 3^{x} \, dx \) using left-hand and right-hand sums with \( n=3 \) subdivisions, we first determine the width of each subdivision. 1. **Calculate the width of each subdivision (Δx):** \[ \Delta x = \frac{b - a}{n} = \frac{8 - (-1)}{3} = \frac{9}{3} = 3 \] 2. **Determine the x-values for the subdivisions:** - For \( n = 3 \), starting from \( x = -1 \): - The left-hand points are: \( x_0 = -1 \), \( x_1 = 2 \), and \( x_2 = 5 \). - The right-hand points are: \( x_1 = 2 \), \( x_2 = 5 \), and \( x_3 = 8 \). 3. **Left-Hand Sum (L):** \[ L = 3^{x_0} \Delta x + 3^{x_1} \Delta x + 3^{x_2} \Delta x = 3^{-1} \cdot 3 + 3^{2} \cdot 3 + 3^{5} \cdot 3 \] Calculate each term: - \( 3^{-1} = \frac{1}{3} \) - \( 3^{2} = 9 \) - \( 3^{5} = 243 \) Now, substituting back: \[ L = \left(\frac{1}{3} \cdot 3\right) + (9 \cdot 3) + (243 \cdot 3) = 1 + 27 + 729 = 757 \] 4. **Right-Hand Sum (R):** \[ R = 3^{x_1} \Delta x + 3^{x_2} \Delta x + 3^{x_3} \Delta x = 3^{2} \cdot 3 + 3^{5} \cdot 3 + 3^{8} \cdot 3 \] We already calculated \( 3^{2} = 9 \) and \( 3^{5} = 243 \). Now, calculate \( 3^{8} \): - \( 3^{8} = 6561 \) Substituting back: \[ R = (9 \cdot 3) + (243 \cdot 3) + (6561 \cdot 3) = 27 + 729 + 19683 = 20439 \] Thus, the left-hand sum is \( L = 757 \) and the right-hand sum is \( R = 20439 \). Left-hand sum \( = 757 \) Right-hand sum \( = 20439 \)