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We are asked to approximate the value of the integral 

\[
\int_0^6 3\sqrt{x}\, dx
\]

using the trapezoidal rule with 4 subintervals for the function \( f(x) = 3\sqrt{x} \).

### Step 1: Define the trapezoidal rule
The trapezoidal rule for \( n \) subintervals is given by:

\[
T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n) \right]
\]

where \( \Delta x = \frac{b-a}{n} \) is the width of each subinterval, and \( x_0, x_1, \dots, x_n \) are the points that divide the interval \([a, b]\) into \(n\) equal parts.

### Step 2: Calculate \( \Delta x \)
Here, the interval is \([0, 6]\) and we are dividing it into 4 subintervals. So,

\[
\Delta x = \frac{6 - 0}{4} = 1.5
\]

### Step 3: Determine the points
The points \( x_0, x_1, x_2, x_3, x_4 \) are:

\[
x_0 = 0, \quad x_1 = 1.5, \quad x_2 = 3, \quad x_3 = 4.5, \quad x_4 = 6
\]

### Step 4: Evaluate the function at each point
Now, we evaluate \( f(x) = 3\sqrt{x} \) at each of these points:

\[
f(x_0) = 3\sqrt{0} = 0
\]
\[
f(x_1) = 3\sqrt{1.5} = 3 \times 1.2247 = 3.6741
\]
\[
f(x_2) = 3\sqrt{3} = 3 \times 1.7321 = 5.1962
\]
\[
f(x_3) = 3\sqrt{4.5} = 3 \times 2.1213 = 6.3639
\]
\[
f(x_4) = 3\sqrt{6} = 3 \times 2.4495 = 7.3485
\]

### Step 5: Apply the trapezoidal rule
Now, apply the trapezoidal rule formula:

\[
T_4 = \frac{1.5}{2} \left[ f(0) + 2f(1.5) + 2f(3) + 2f(4.5) + f(6) \right]
\]

Substitute the values:

\[
T_4 = \frac{1.5}{2} \left[ 0 + 2(3.6741) + 2(5.1962) + 2(6.3639) + 7.3485 \right]
\]

\[
T_4 = \frac{1.5}{2} \left[ 0 + 7.3482 + 10.3924 + 12.7278 + 7.3485 \right]
\]

\[
T_4 = \frac{1.5}{2} \times 37.8169
\]

\[
T_4 = 0.75 \times 37.8169 = 28.3627
\]

### Final Answer:
The value of the trapezoidal sum approximation is approximately **28.363** (rounded to the nearest thousandth).

Would you like to go over any details or explanations?

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Here are 5 related questions you could explore:
1. How does the trapezoidal rule compare with Simpson's rule for approximating integrals?
2. What is the geometric interpretation of the trapezoidal rule?
3. How does increasing the number of subintervals affect the accuracy of the trapezoidal rule?
4. Can we find the exact value of the integral \( \int_0^6 3\sqrt{x} \, dx \) and compare it with the approximation?
5. How would the approximation change if we used more subintervals?

**Tip**: The more subintervals you use in the trapezoidal rule, the more accurate your approximation will be!

If four subintervals of equal length are used, what is the value of the trapezoidal sum approximation for the integral of 3√x from 0 to 6? Round to the nearest thousandth if necessary.

This example demonstrates how to approximate the integral of 3√x from 0 to 6 using the trapezoidal rule with four equal subintervals. Learn the step-by-step solution using calculus and numerical integration concepts.

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