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 \( = \)

To estimate the integral \( \int_{1}^{7} \sqrt{x} \, dx \) using a left-hand sum and a right-hand sum with \( n=3 \) subdivisions, we start by determining the width of each subdivision. The endpoint values are \( a = 1 \) and \( b = 7 \). The width \( \Delta x \) for each subdivision is given by: \[ \Delta x = \frac{b-a}{n} = \frac{7 - 1}{3} = \frac{6}{3} = 2. \] Now, we'll find the left-hand sum. The left-hand endpoints for the 3 subdivisions are 1, 3, and 5. The function values at these points are: - \( f(1) = \sqrt{1} = 1 \) - \( f(3) = \sqrt{3} \approx 1.732 \) - \( f(5) = \sqrt{5} \approx 2.236 \) The left-hand sum \( L \) is: \[ L = \Delta x (f(1) + f(3) + f(5)) = 2 (1 + \sqrt{3} + \sqrt{5}) \approx 2(1 + 1.732 + 2.236) = 2 \times 5.968 \approx 11.936. \] So, rounding to two decimal places, the left-hand sum is: Left-hand sum \( = 11.94 \). Now, for the right-hand sum, the right-hand endpoints are at 3, 5, and 7. The function values at these points are: - \( f(3) = \sqrt{3} \approx 1.732 \) - \( f(5) = \sqrt{5} \approx 2.236 \) - \( f(7) = \sqrt{7} \approx 2.646 \) The right-hand sum \( R \) is: \[ R = \Delta x (f(3) + f(5) + f(7)) = 2 (\sqrt{3} + \sqrt{5} + \sqrt{7}) \approx 2 (1.732 + 2.236 + 2.646) = 2 \times 6.614 \approx 13.228. \] So, rounding to two decimal places, the right-hand sum is: Right-hand sum \( = 13.23 \).