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We are asked to approximate the integral \(\int_1^9 5\sqrt{x} \, dx\) using the midpoint Riemann sum with 4 subintervals of equal length.

### Step 1: Define the function and interval
The function given is:
\[
f(x) = 5\sqrt{x}
\]
The interval is \([1, 9]\).

### Step 2: Determine the width of each subinterval
The total interval length is \(9 - 1 = 8\). Since we are dividing the interval into 4 subintervals, the width of each subinterval, \(\Delta x\), is:
\[
\Delta x = \frac{8}{4} = 2
\]

### Step 3: Identify the midpoints of each subinterval
The midpoints of the subintervals are found by taking the average of the endpoints of each subinterval. The subintervals are:
- \([1, 3]\)
- \([3, 5]\)
- \([5, 7]\)
- \([7, 9]\)

The midpoints are:
- \(x_1 = \frac{1 + 3}{2} = 2\)
- \(x_2 = \frac{3 + 5}{2} = 4\)
- \(x_3 = \frac{5 + 7}{2} = 6\)
- \(x_4 = \frac{7 + 9}{2} = 8\)

### Step 4: Calculate the function values at the midpoints
We now evaluate \(f(x)\) at each midpoint:
\[
f(2) = 5\sqrt{2}, \quad f(4) = 5\sqrt{4}, \quad f(6) = 5\sqrt{6}, \quad f(8) = 5\sqrt{8}
\]

These simplify to:
\[
f(2) = 5\sqrt{2}, \quad f(4) = 10, \quad f(6) = 5\sqrt{6}, \quad f(8) = 10\sqrt{2}
\]

### Step 5: Calculate the Riemann sum
The midpoint Riemann sum is given by:
\[
\text{Riemann sum} = \Delta x \left( f(x_1) + f(x_2) + f(x_3) + f(x_4) \right)
\]

Substituting the values:
\[
\text{Riemann sum} = 2 \left( 5\sqrt{2} + 10 + 5\sqrt{6} + 10\sqrt{2} \right)
\]
Let's calculate the approximate numerical value of this expression.

I will compute this now.The value of the midpoint Riemann sum approximation for the integral \(\int_1^9 5\sqrt{x} \, dx\), using four subintervals, is approximately **86.921** when rounded to the nearest thousandth.

Would you like further details or clarifications? Here are some related questions for deeper understanding:

1. How does the midpoint Riemann sum differ from left or right Riemann sums?
2. What are the steps to calculate the definite integral exactly instead of approximating?
3. How does the accuracy of the approximation change with more subintervals?
4. Can the midpoint rule be applied to functions that are not continuous?
5. What other methods, besides Riemann sums, are commonly used for approximating integrals?

**Tip:** Increasing the number of subintervals generally improves the accuracy of Riemann sum approximations!

Let f be the function defined by f(x) = 5√x. If four subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for ∫₁⁹ 5√x dx? Round to the nearest thousandth if necessary.

Calculate the midpoint Riemann sum approximation for the integral ∫₁⁹ 5√x dx, using four subintervals of equal length. This problem covers integral approximation concepts such as the midpoint rule, function evaluation, and square roots. The final answer is rounded to the nearest thousandth.

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