Question Find the relative error when using the midpoint rule to calculate \( \int_{0}^{2}\left(3 x^{2}+2\right) d x \) using 4 subintervals. Find the relative error as a percent rounded to the nearest hundredth if necessary. Do not round until your final answer and do not inclu the symbol "\%" in your answer.

To find the relative error when using the midpoint rule to calculate the integral \( \int_{0}^{2}\left(3 x^{2}+2\right) d x \) using 4 subintervals, we will follow these steps: 1. **Calculate the exact value of the integral.** 2. **Apply the midpoint rule to estimate the integral.** 3. **Calculate the relative error.** ### Step 1: Calculate the Exact Value of the Integral The exact value of the integral can be calculated as follows: \[ \int_{0}^{2} (3x^2 + 2) \, dx \] First, we find the antiderivative: \[ F(x) = x^3 + 2x \] Now, we evaluate it from 0 to 2: \[ F(2) - F(0) = (2^3 + 2 \cdot 2) - (0^3 + 2 \cdot 0) = (8 + 4) - 0 = 12 \] So, the exact value of the integral is \( 12 \). ### Step 2: Apply the Midpoint Rule We will divide the interval \([0, 2]\) into 4 subintervals. The width of each subinterval \( \Delta x \) is: \[ \Delta x = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5 \] The midpoints of each subinterval are: - For the first subinterval \([0, 0.5]\): midpoint \( x_1 = 0.25 \) - For the second subinterval \([0.5, 1]\): midpoint \( x_2 = 0.75 \) - For the third subinterval \([1, 1.5]\): midpoint \( x_3 = 1.25 \) - For the fourth subinterval \([1.5, 2]\): midpoint \( x_4 = 1.75 \) Now we evaluate the function at each midpoint: \[ f(0.25) = 3(0.25)^2 + 2 = 3(0.0625) + 2 = 0.1875 + 2 = 2.1875 \] \[ f(0.75) = 3(0.75)^2 + 2 = 3(0.5625) + 2 = 1.6875 + 2 = 3.6875 \] \[ f(1.25) = 3(1.25)^2 + 2 = 3(1.5625) + 2 = 4.6875 + 2 = 6.6875 \] \[ f(1.75) = 3(1.75)^2 + 2 = 3(3.0625) + 2 = 9.1875 + 2 = 11.1875 \] Now we apply the midpoint rule: \[ \text{Midpoint Rule Estimate} = \Delta x \left( f(x_1) + f(x_2) + f(x_3) + f(x_4) \right) \] \[ = 0.5 \left( 2.1875 + 3.6875 + 6.6875 + 11.1875 \right) \] \[ = 0.5 \left( 23.75 \right) = 11.875 \] ### Step 3: Calculate the Relative Error The relative error is given by: \[ \text{Relative Error} = \left| \frac{\text{Exact Value} - \text{Estimate}}{\text{Exact Value}} \right| \] Substituting the values we found: \[ \text{Relative Error} = \left| \frac{12 - 11.875}{12} \right| = \left| \frac{0.125}{12} \right| = \frac{0.125}{12} \approx 0.01041667 \] To express this as a percentage, we multiply by 100: \[ \text{Relative Error as a percent} = 0.01041667 \times 100 \approx 1.04166667 \] Rounding to the nearest hundredth: \[ \text{Final Answer} = 1.04 \] Thus, the relative error as a percent rounded to the nearest hundredth is \( 1.04 \).