Determine the absolute error when using the midpoint rule to find $\int_{1}^{3}\left(8 x^{2}+3\right) d x$ using 4 subintervals. Enter an exact value. Do not enter the answer as a percent.
Answer
Final Answer: The absolute error when using the midpoint rule to find \(\int_{1}^{3}\left(8 x^{2}+3\right) d x\) using 4 subintervals is \(\boxed{0.333333333333329}\).
Step 1 ：Define the function \(f = 8x^2 + 3\) and the variable \(x\).
Step 2 ：Calculate the exact value of the integral \(\int_{1}^{3} f dx\), which is \(\frac{226}{3}\).
Step 3 ：Calculate the approximate value of the integral using the midpoint rule with 4 subintervals. The width of each subinterval \(h\) is \(0.5\), and the midpoints are \(1.25, 1.75, 2.25, 2.75\). The approximate value is \(75\).
Step 4 ：Calculate the absolute error, which is the absolute difference between the exact value and the approximate value. The absolute error is \(0.333333333333329\).
Step 5 ：Final Answer: The absolute error when using the midpoint rule to find \(\int_{1}^{3}\left(8 x^{2}+3\right) d x\) using 4 subintervals is \(\boxed{0.333333333333329}\).
