Questions: Estimate the definite integral ∫ from 3 to 4 of √x dx using a left Riemann sum. Complete the table below by filling in the left Riemann sum of f(x)=√x over the interval [3,4] with n=4 subdivisions. Then estimate the sum. (Do not round your answers in the table, but round the sum to 2 decimal places.) Delta x x f(x) Area 0.25 3 √3 (0.25)(√3) 0.25 3.25 0.25 3.5 √3.5 0.25 √3.75 (√3.25) Sum:

Solution Steps

To estimate the definite integral using a left Riemann sum, we need to:

1. Divide the interval \([3, 4]\) into \(n = 4\) equal subdivisions.
2. Calculate the width of each subdivision, \(\Delta x = \frac{4-3}{4} = 0.25\).
3. Determine the left endpoints of each subdivision.
4. Evaluate the function \(f(x) = \sqrt{x}\) at each left endpoint.
5. Multiply each function value by \(\Delta x\) to find the area of each rectangle.
6. Sum these areas to get the left Riemann sum.

The interval is \([3, 4]\) and we divide it into \(n = 4\) equal subdivisions. The width of each subdivision is: \[ \Delta x = \frac{4 - 3}{4} = 0.25 \]

The left endpoints of each subdivision are: \[ x_0 = 3, \quad x_1 = 3.25, \quad x_2 = 3.5, \quad x_3 = 3.75 \]

The function \(f(x) = \sqrt{x}\) evaluated at each left endpoint is: \[ f(3) = \sqrt{3} \approx 1.7321 \] \[ f(3.25) = \sqrt{3.25} \approx 1.8028 \] \[ f(3.5) = \sqrt{3.5} \approx 1.8708 \] \[ f(3.75) = \sqrt{3.75} \approx 1.9365 \]

The area of each rectangle is given by \(\Delta x \cdot f(x_i)\): \[ \text{Area}_0 = 0.25 \cdot \sqrt{3} \approx 0.4330 \] \[ \text{Area}_1 = 0.25 \cdot \sqrt{3.25} \approx 0.4507 \] \[ \text{Area}_2 = 0.25 \cdot \sqrt{3.5} \approx 0.4677 \] \[ \text{Area}_3 = 0.25 \cdot \sqrt{3.75} \approx 0.4841 \]

The left Riemann sum is the sum of the areas of the rectangles: \[ \text{Sum} = 0.4330 + 0.4507 + 0.4677 + 0.4841 \approx 1.8355 \]

Final Answer