Use the trapezoidal rule to
approximate the integral with n
trapezoids. Round your answer
to the nearest thousandth.

Ask by Robbins Willis. in the United States

Mar 22,2025

Question

Use the trapezoidal rule to approximate the integral with n trapezoids. Round your answer to the nearest thousandth. $$ \int_{2}^{8} \sqrt{x^{3}+4} \mathrm{dx} \approx[?] \quad \mathrm{n}=3 $$

UpStudy AI · Accepted Answer

To approximate the integral $$ \int_{2}^{8} \sqrt{x^{3}+4} \, dx $$ using the trapezoidal rule with $$ n = 3 $$, we will follow these steps:

1. **Determine the width of each trapezoid**:
   $$
   h = \frac{b - a}{n}
   $$
   where $$ a = 2 $$, $$ b = 8 $$, and $$ n = 3 $$.

2. **Calculate the x-values at which we will evaluate the function**:
   $$
   x_0 = a, \quad x_1 = a + h, \quad x_2 = a + 2h, \quad x_3 = b
   $$

3. **Evaluate the function at these x-values**:
   $$
   f(x) = \sqrt{x^{3} + 4}
   $$

4. **Apply the trapezoidal rule formula**:
   $$
   \text{Approximation} = \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3) \right)
   $$

Now, let's perform the calculations step by step.

### Step 1: Calculate $$ h $$
$$
h = \frac{8 - 2}{3}
$$

### Step 2: Calculate the x-values
$$
x_0 = 2, \quad x_1 = 2 + h, \quad x_2 = 2 + 2h, \quad x_3 = 8
$$

### Step 3: Evaluate the function at these x-values
$$
f(x) = \sqrt{x^{3} + 4}
$$

Now, I will calculate $$ h $$, the x-values, and the function evaluations.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{8^{3}+4}$$
- step1: Add the numbers:
  $$\sqrt{516}$$
- step2: Write as a product:
  $$\sqrt{4\times 129}$$
- step3: Write in exponential form:
  $$\sqrt{2^{2}\times 129}$$
- step4: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{129}$$
- step5: Simplify the root:
  $$2\sqrt{129}$$
Calculate or simplify the expression $$ 2 + 2 * (8 - 2) / 3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2+\frac{2\left(8-2\right)}{3}$$
- step1: Subtract the numbers:
  $$2+\frac{2\times 6}{3}$$
- step2: Reduce the fraction:
  $$2+4$$
- step3: Add the numbers:
  $$6$$
Calculate or simplify the expression $$ \sqrt((2 + (8 - 2) / 3)^3 + 4) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\left(2+\frac{\left(8-2\right)}{3}\right)^{3}+4}$$
- step1: Remove the parentheses:
  $$\sqrt{\left(2+\frac{8-2}{3}\right)^{3}+4}$$
- step2: Subtract the numbers:
  $$\sqrt{\left(2+\frac{6}{3}\right)^{3}+4}$$
- step3: Divide the terms:
  $$\sqrt{\left(2+2\right)^{3}+4}$$
- step4: Add the numbers:
  $$\sqrt{4^{3}+4}$$
- step5: Add the numbers:
  $$\sqrt{68}$$
- step6: Write as a product:
  $$\sqrt{4\times 17}$$
- step7: Write in exponential form:
  $$\sqrt{2^{2}\times 17}$$
- step8: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{17}$$
- step9: Simplify the root:
  $$2\sqrt{17}$$
Calculate or simplify the expression $$ \sqrt(2^3 + 4) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{2^{3}+4}$$
- step1: Add the numbers:
  $$\sqrt{12}$$
- step2: Write as a product:
  $$\sqrt{4\times 3}$$
- step3: Write in exponential form:
  $$\sqrt{2^{2}\times 3}$$
- step4: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{3}$$
- step5: Simplify the root:
  $$2\sqrt{3}$$
Calculate or simplify the expression $$ (8 - 2) / 3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(8-2\right)}{3}$$
- step1: Remove the parentheses:
  $$\frac{8-2}{3}$$
- step2: Subtract the numbers:
  $$\frac{6}{3}$$
- step3: Reduce the numbers:
  $$\frac{2}{1}$$
- step4: Calculate:
  $$2$$
Calculate or simplify the expression $$ \sqrt((2 + 2 * (8 - 2) / 3)^3 + 4) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\left(2+\frac{2\left(8-2\right)}{3}\right)^{3}+4}$$
- step1: Subtract the numbers:
  $$\sqrt{\left(2+\frac{2\times 6}{3}\right)^{3}+4}$$
- step2: Reduce the fraction:
  $$\sqrt{\left(2+4\right)^{3}+4}$$
- step3: Add the numbers:
  $$\sqrt{6^{3}+4}$$
- step4: Add the numbers:
  $$\sqrt{220}$$
- step5: Write as a product:
  $$\sqrt{4\times 55}$$
- step6: Write in exponential form:
  $$\sqrt{2^{2}\times 55}$$
- step7: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{55}$$
- step8: Simplify the root:
  $$2\sqrt{55}$$
Calculate or simplify the expression $$ 2 + (8 - 2) / 3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2+\frac{\left(8-2\right)}{3}$$
- step1: Remove the parentheses:
  $$2+\frac{8-2}{3}$$
- step2: Subtract the numbers:
  $$2+\frac{6}{3}$$
- step3: Divide the terms:
  $$2+2$$
- step4: Add the numbers:
  $$4$$
### Step 1: Calculate $$ h $$
$$
h = \frac{8 - 2}{3} = 2
$$

### Step 2: Calculate the x-values
$$
x_0 = 2, \quad x_1 = 4, \quad x_2 = 6, \quad x_3 = 8
$$

### Step 3: Evaluate the function at these x-values
- $$ f(x_0) = f(2) = \sqrt{2^3 + 4} = \sqrt{8 + 4} = \sqrt{12} \approx 3.464 $$
- $$ f(x_1) = f(4) = \sqrt{4^3 + 4} = \sqrt{64 + 4} = \sqrt{68} \approx 8.246 $$
- $$ f(x_2) = f(6) = \sqrt{6^3 + 4} = \sqrt{216 + 4} = \sqrt{220} \approx 14.832 $$
- $$ f(x_3) = f(8) = \sqrt{8^3 + 4} = \sqrt{512 + 4} = \sqrt{516} \approx 22.716 $$

### Step 4: Apply the trapezoidal rule formula
Now we can substitute these values into the trapezoidal rule formula:
$$
\text{Approximation} = \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3) \right)
$$
Substituting the values:
$$
\text{Approximation} = \frac{2}{2} \left( 3.464 + 2 \cdot 8.246 + 2 \cdot 14.832 + 22.716 \right)
$$

Now, let's calculate this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$1\times \left(3.464+2\times 8.246+2\times 14.832+22.716\right)$$
- step1: Multiply the numbers:
  $$1\times \left(3.464+16.492+2\times 14.832+22.716\right)$$
- step2: Multiply the numbers:
  $$1\times \left(3.464+16.492+29.664+22.716\right)$$
- step3: Add the numbers:
  $$1\times 72.336$$
- step4: Multiply:
  $$72.336$$
The approximation using the trapezoidal rule is:
$$
\text{Approximation} \approx 72.336
$$

Rounding this to the nearest thousandth, we get:
$$
\text{Approximation} \approx 72.336
$$

Thus, the final answer is:
$$
\int_{2}^{8} \sqrt{x^{3}+4} \, dx \approx 72.336
$$
The integral $$ \int_{2}^{8} \sqrt{x^{3}+4} \, dx $$ is approximately 72.336 when using the trapezoidal rule with 3 trapezoids.

Use the trapezoidal rule to
approximate the integral with n
trapezoids. Round your answer
to the nearest thousandth.

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