Question

Multiple Choice If three equal subdivisions of \([-2,4]\) are used, what is the trapezoidal approximation of \(\int_{-2}^{4} \frac{e^{x}}{2} d x ?\) \begin{array}{l}{\text { (A) } e^{4}+e^{2}+e^{0}+e^{-2}} \\ {\text { (B) } e^{4}+2 e^{2}+2 e^{0}+e^{-2}} \\ {\text { (C) } \frac{1}{2}\left(e^{4}+e^{2}+e^{0}+e^{-2}\right)} \\ {\text { (D) } \frac{1}{2}\left(e^{4}+2 e^{2}+2 e^{0}+e^{-2}\right)} \\ {\text { (E) } \frac{1}{4}\left(e^{4}+2 e^{2}+2 e^{0}+e^{-2}\right)}\end{array}

Short answer

(D) \( \frac{1}{2}\left(e^{4}+2 e^{2}+2 e^{0}+e^{-2}\right) \)

Step-by-step solution

Identify the Subdivisions of the Interval

The integral is calculated from \(-2\) to \(4\) and we are told to divide this interval into three equal parts. This means each subdivision (or 'step') is \(\frac{4 - (-2)}{3}= 2\). Therefore, our x-values or subdivisions are -2, 0, 2 and 4.

Evaluate the Function at the Subdivisions

The function here is \(\frac{e^{x}}{2}\). We need to evaluate this function at the identified x-values. This gives: \(\frac{e^{-2}}{2}\), \(\frac{e^{0}}{2}\), \(\frac{e^{2}}{2}\), and \(\frac{e^{4}}{2}\).

Apply the Trapezoidal Rule

Now, plug these values into the trapezoidal rule formula: \(h = 2\), \(x_{0} = -2\), \(x_{1} = 0\), \(x_{2} = 2\), \(x_{3} = 4\). So the trapezoidal approximation is: \(\frac{2}{2}\left[\frac{e^{-2}}{2} + 2\left(\frac{e^{0}}{2}\right) + 2\left(\frac{e^{2}}{2}\right) + \frac{e^{4}}{2}\right] \) = \(\frac{1}{2}\left(e^{4}+2 e^{2}+2 e^{0}+e^{-2}\right) \)

Compare with the Provided Choices

Now that we have our numerical approximation, we should compare it to the answer choices on the multiple choice question. It matches choice (D) exactly, so that is our answer.