Questions: The instructions for the given integral have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson's Rule. Complete the following part: integral from -1 to 1 of (3x^2 + 1) dx (Round to two decimal places as needed.) b. Evaluate the integral directly and find EM. integral from -1 to 1 of (3x^2 + 1) dx = 4.00 (Round to two decimal places as needed.) EM = (Round to two decimal places as needed.)

The instructions for the given integral have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson's Rule. Complete the following part: integral from -1 to 1 of (3x^2 + 1) dx (Round to two decimal places as needed.) b. Evaluate the integral directly and find EM. integral from -1 to 1 of (3x^2 + 1) dx = 4.00 (Round to two decimal places as needed.) EM = (Round to two decimal places as needed.)
Transcript text: The instructions for the given integral have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson's Rule. Complete the following par \[ \int_{-1}^{1}\left(3 x^{2}+1\right) d x \] (Round to two decimal places as needed.) b. Evaluate the integral directly and find $\left|E_{M}\right|$. \[ \int_{-1}^{1}\left(3 x^{2}+1\right) d x=4.00 \] (Round to two decimal places as needed.) \[ \left|E_{M}\right|=\square \] (Round to two decimal places as needed.)
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Solution

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Solution Steps

To solve the given integral and find the error \( \left|E_{M}\right| \) using the Midpoint Rule, we need to follow these steps:

  1. Evaluate the integral directly to get the exact value.
  2. Use the Midpoint Rule to approximate the integral.
  3. Calculate the absolute error \( \left|E_{M}\right| \) by taking the absolute difference between the exact value and the Midpoint Rule approximation.
Solution Approach
  1. Evaluate the integral directly.
  2. Use the Midpoint Rule to approximate the integral.
  3. Calculate the absolute error \( \left|E_{M}\right| \).
Step 1: Evaluate the Integral Directly

To evaluate the integral \( \int_{-1}^{1} (3x^2 + 1) \, dx \), we compute:

\[ \int (3x^2 + 1) \, dx = x^3 + x + C \]

Evaluating from \( -1 \) to \( 1 \):

\[ \left[ x^3 + x \right]_{-1}^{1} = (1^3 + 1) - ((-1)^3 + (-1)) = (1 + 1) - (-1 - 1) = 2 - (-2) = 4 \]

Thus, the exact value of the integral is:

\[ \int_{-1}^{1} (3x^2 + 1) \, dx = 4.00 \]

Step 2: Midpoint Rule Approximation

Using the Midpoint Rule with \( n = 100 \) subintervals, we calculate the width of each subinterval:

\[ h = \frac{b - a}{n} = \frac{1 - (-1)}{100} = 0.02 \]

The midpoints are given by:

\[ \text{midpoints} = \left[-1 + \frac{h}{2}, -1 + \frac{3h}{2}, \ldots, 1 - \frac{h}{2}\right] \]

The Midpoint Rule approximation is:

\[ \text{Midpoint Approximation} = h \sum_{i=1}^{n} f\left(a + (i - 0.5)h\right) \]

Calculating this yields:

\[ \text{Midpoint Approximation} \approx 3.9998 \]

Step 3: Calculate the Absolute Error

The absolute error \( |E_M| \) is calculated as follows:

\[ |E_M| = \left| \text{Exact Value} - \text{Midpoint Approximation} \right| = |4.00 - 3.9998| = 0.0002 \]

Final Answer

The results are summarized as follows:

  • Exact value of the integral: \( 4.00 \)
  • Midpoint Rule approximation: \( 4.00 \)
  • Absolute error \( |E_M| \): \( 0.00 \)

Thus, the final answers are:

\[ \boxed{4.00} \] \[ \boxed{0.00} \]

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