Approximate the integral below using a left hand sum with 5 rectangles. Round your final answer to the nearest thousandth. \( \int_{4}^{9}\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right) d x \approx[?] \)

We are given the integral \[ \int_{4}^{9} \left( \frac{1-\sqrt{x}}{\sqrt{x}} \right) dx \] and we need to approximate its value using a left-hand sum with 5 rectangles. **Step 1. Determine the width of each rectangle** The interval is from \( x=4 \) to \( x=9 \). The width \( \Delta x \) is given by \[ \Delta x = \frac{9-4}{5} = 1. \] **Step 2. Identify the left endpoints** For a left-hand sum with 5 rectangles, the left endpoints are: \[ x_0 = 4,\quad x_1 = 5,\quad x_2 = 6,\quad x_3 = 7,\quad x_4 = 8. \] **Step 3. Write the function in a simpler form** The integrand is \[ \frac{1-\sqrt{x}}{\sqrt{x}} = \frac{1}{\sqrt{x}} - 1. \] **Step 4. Evaluate the function at each left endpoint** 1. For \( x=4 \): \[ f(4) = \frac{1}{\sqrt{4}} - 1 = \frac{1}{2} - 1 = -0.5. \] 2. For \( x=5 \): \[ \sqrt{5} \approx 2.23607,\quad f(5) \approx \frac{1}{2.23607} - 1 \approx 0.4472 - 1 = -0.5528. \] 3. For \( x=6 \): \[ \sqrt{6} \approx 2.44949,\quad f(6) \approx \frac{1}{2.44949} - 1 \approx 0.4082 - 1 = -0.5918. \] 4. For \( x=7 \): \[ \sqrt{7} \approx 2.64575,\quad f(7) \approx \frac{1}{2.64575} - 1 \approx 0.3780 - 1 = -0.6220. \] 5. For \( x=8 \): \[ \sqrt{8} \approx 2.82843,\quad f(8) \approx \frac{1}{2.82843} - 1 \approx 0.3536 - 1 = -0.6464. \] **Step 5. Compute the left-hand sum** The left-hand sum is given by \[ L = \Delta x \left[f(4) + f(5) + f(6) + f(7) + f(8)\right]. \] Substitute the values: \[ \begin{aligned} L &\approx 1 \times \Big[ -0.5 - 0.5528 - 0.5918 - 0.6220 - 0.6464 \Big] \\ &\approx - (0.5 + 0.5528 + 0.5918 + 0.6220 + 0.6464) \\ &\approx -2.9130. \end{aligned} \] Rounding to the nearest thousandth, we have \[ L \approx -2.913. \] **Final Answer:** \[ \boxed{-2.913} \]