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

Ask by Hodges Bowers. 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.

Ask by Hodges Bowers. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

To approximate the integral $$ \int_{2}^{8} \sqrt{x^{3}+4} \, dx $$ using the trapezoidal rule with $$ n = 3 $$ trapezoids, 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 the endpoints and midpoints**:
   - $$ x_0 = a $$
   - $$ x_1 = a + h $$
   - $$ x_2 = a + 2h $$
   - $$ x_3 = b $$

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

4. **Apply the trapezoidal rule formula**:
   $$
   \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3) \right)
   $$

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

### Step 1: Calculate $$ h $$
$$
h = \frac{8 - 2}{3} = \frac{6}{3} = 2
$$

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

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

### Step 4: Apply the trapezoidal rule formula
Now we can substitute these values into the trapezoidal rule formula.

Let's calculate the values of $$ f(x_0) $$, $$ f(x_1) $$, $$ f(x_2) $$, and $$ f(x_3) $$ first, and then apply the trapezoidal rule.

I'll compute these values now.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{220}$$
- step1: Write as a product:
  $$\sqrt{4\times 55}$$
- step2: Write in exponential form:
  $$\sqrt{2^{2}\times 55}$$
- step3: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{55}$$
- step4: Simplify the root:
  $$2\sqrt{55}$$
Calculate or simplify the expression $$ \sqrt(516) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{516}$$
- step1: Write as a product:
  $$\sqrt{4\times 129}$$
- step2: Write in exponential form:
  $$\sqrt{2^{2}\times 129}$$
- step3: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{129}$$
- step4: Simplify the root:
  $$2\sqrt{129}$$
Calculate or simplify the expression $$ \sqrt(68) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{68}$$
- step1: Write as a product:
  $$\sqrt{4\times 17}$$
- step2: Write in exponential form:
  $$\sqrt{2^{2}\times 17}$$
- step3: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{17}$$
- step4: Simplify the root:
  $$2\sqrt{17}$$
Calculate or simplify the expression $$ \sqrt(12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{12}$$
- step1: Write as a product:
  $$\sqrt{4\times 3}$$
- step2: Write in exponential form:
  $$\sqrt{2^{2}\times 3}$$
- step3: Use the properties of radicals:
  $$\sqrt{2^{2}}\times \sqrt{3}$$
- step4: Simplify the root:
  $$2\sqrt{3}$$
The evaluated function values are as follows:

- $$ f(x_0) = f(2) \approx 3.464 $$
- $$ f(x_1) = f(4) \approx 8.246 $$
- $$ f(x_2) = f(6) \approx 14.832 $$
- $$ f(x_3) = f(8) \approx 22.716 $$

### Step 4: Apply the trapezoidal rule formula
Now we can substitute these values into the trapezoidal rule formula:

$$
\int_{2}^{8} f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3) \right)
$$

Substituting $$ h = 2 $$:

$$
\int_{2}^{8} f(x) \, dx \approx \frac{2}{2} \left( 3.464 + 2 \cdot 8.246 + 2 \cdot 14.832 + 22.716 \right)
$$

This simplifies to:

$$
\int_{2}^{8} f(x) \, dx \approx 1 \left( 3.464 + 16.492 + 29.664 + 22.716 \right)
$$

Now, let's calculate the sum inside the parentheses.
Calculate the value by following steps:
- step0: Calculate:
  $$3.464+16.492+29.664+22.716$$
- step1: Add the numbers:
  $$72.336$$
The sum inside the parentheses is:

$$
3.464 + 16.492 + 29.664 + 22.716 \approx 72.336
$$

### Final Calculation
Now, substituting this back into the trapezoidal rule formula:

$$
\int_{2}^{8} f(x) \, dx \approx 1 \cdot 72.336 = 72.336
$$

Rounding to the nearest thousandth, the approximate value of the integral is:

$$
\int_{2}^{8} \sqrt{x^{3}+4} \, dx \approx 72.336
$$

Thus, the final answer is $$ \approx 72.336 $$.
The approximate value of the integral is $$ \approx 72.336 $$.

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Calculate

Step 2: Calculate the x-values

Step 3: Evaluate the function at these x-values

Step 4: Apply the trapezoidal rule formula

Step 4: Apply the trapezoidal rule formula

Final Calculation

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Calculate

Step 2: Calculate the x-values

Step 3: Evaluate the function at these x-values

Step 4: Apply the trapezoidal rule formula

Step 4: Apply the trapezoidal rule formula

Final Calculation

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions

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