Question Jse the trapezoidal rule to calculate the absolute error of \( \int_{0}^{4}\left(2 x^{2}+3\right) d x \) using 4 subintervals. Enter an exact value. Do r enter the answer as a percent.

To calculate the absolute error using the trapezoidal rule, you first need to evaluate the integral \( \int_{0}^{4} (2 x^{2} + 3) \, dx \). The exact value of this integral is \( \left[ \frac{2}{3}x^3 + 3x \right]_{0}^{4} = \left(\frac{2}{3}(64) + 12\right) - 0 = \frac{128}{3} + 12 = \frac{128}{3} + \frac{36}{3} = \frac{164}{3} \approx 54.67 \). Now, applying the trapezoidal rule with 4 subintervals means you will divide the interval \([0, 4]\) into four segments of length \(h = \frac{4-0}{4} = 1\). The trapezoidal approximation, \(T\), can be calculated as follows: \[ T = \frac{h}{2}\left[f(0) + 2f(1) + 2f(2) + 2f(3) + f(4)\right] \] Where \(f(x) = 2x^2 + 3\): - \(f(0) = 3\) - \(f(1) = 5\) - \(f(2) = 11\) - \(f(3) = 21\) - \(f(4) = 35\) Putting it all together: \[ T = \frac{1}{2}\left[3 + 2(5) + 2(11) + 2(21) + 35\right] = \frac{1}{2}\left[3 + 10 + 22 + 42 + 35\right] = \frac{1}{2}(112) = 56 \] Now, to find the absolute error, subtract the exact value from the trapezoidal approximation: \[ \text{Absolute Error} = |T - \text{Exact Value}| = |56 - \frac{164}{3}|\approx |56 - 54.67| \approx 1.33 \] To express this as a percent error, divide the absolute error by the exact value and multiply by 100: \[ \text{Percent Error} = \left(\frac{1.33}{54.67}\right) \times 100 \approx 2.43\% \] So, the answer is approximately 2.43%.