Give a 4-term left Riemann sum approximation for the integral below. ∫^14 10 −4x+6‾‾‾‾‾√dx

Answer 1

To find a 4-term left Riemann sum approximation for the integral ∫(10 - 4x + 6√x)dx from 10 to 14, we'll divide the interval [10, 14] into four subintervals of equal width and use the left endpoint of each subinterval to approximate the integral.

First, let's determine the width of each subinterval:

Width of each subinterval = (b - a) / n

where:

- a is the lower limit of integration (10)

- b is the upper limit of integration (14)

- n is the number of subintervals (4)

Width of each subinterval = (14 - 10) / 4 = 4 / 4 = 1

Now, we'll calculate the left Riemann sum using four subintervals:

Left Riemann sum ≈ Δx [f(a) + f(a + Δx) + f(a + 2Δx) + f(a + 3Δx)]

Where:

- Δx is the width of each subinterval (1)

- f(x) is the function we're integrating (10 - 4x + 6√x)

Let's calculate it step by step:

1. Left Riemann sum ≈ 1 [f(10) + f(11) + f(12) + f(13)]

2. Evaluate the function f(x) at each left endpoint:

- f(10) = 10 - 4(10) + 6√10

- f(11) = 10 - 4(11) + 6√11

- f(12) = 10 - 4(12) + 6√12

- f(13) = 10 - 4(13) + 6√13

3. Perform the calculations:

- f(10) = 10 - 40 + 6√10 = -30 + 6√10

- f(11) = 10 - 44 + 6√11 = -34 + 6√11

- f(12) = 10 - 48 + 6√12 = -38 + 6√12

- f(13) = 10 - 52 + 6√13 = -42 + 6√13

4. Plug these values into the left Riemann sum formula:

Left Riemann sum ≈ 1 [(-30 + 6√10) + (-34 + 6√11) + (-38 + 6√12) + (-42 + 6√13)]

5. Calculate the sum:

Left Riemann sum ≈ 1 [-30 - 34 - 38 - 42 + 6(√10 + √11 + √12 + √13)]

6. Simplify:

Left Riemann sum ≈ -144 + 6(√10 + √11 + √12 + √13)

This is the 4-term left Riemann sum approximation for the given integral ∫(10 - 4x + 6√x)dx from 10 to 14.