Let's solve the problem **step-by-step** as requested: --- ### **Step 1: Understanding the Integral and Its Approximation** The given integral is: \[ \int_{-3}^{3} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2} t^2\right] dt \] This is the integral of the standard normal (Gaussian) probability density function from \(-3\) to \(3\). We are asked to approximate this integral using a **Riemann sum**: \[ \sum_{n=-L}^{L} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2}(n\Delta)^2\right] \Delta \] where \(L\) is the integer closest to \(3/\Delta\). **Explanation:** This sum replaces the continuous variable \(t\) with the discrete variable \(n\Delta\), and the integration step \(dt\) with \(\Delta\), essentially approximating the area under the curve with rectangles. --- ### **Step 2: Calculate the Approximation for Different Δ** Let’s compute \(L\) and write the sum for each given \(\Delta\): #### **a) For \(\Delta = 0.1\):** - \(L =\) integer closest to \(3/0.1 = 30\) - So, \(L = 30\) The sum: \[ \sum_{n=-30}^{30} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2}(n \times 0.1)^2\right] \times 0.1 \] #### **b) For \(\Delta = 0.01\):** - \(L =\) integer closest to \(3/0.01 = 300\) - So, \(L = 300\) The sum: \[ \sum_{n=-300}^{300} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2}(n \times 0.01)^2\right] \times 0.01 \] #### **c) For \(\Delta = 0.001\):** - \(L =\) integer closest to \(3/0.001 = 3000\) - So, \(L = 3000\) The sum: \[ \sum_{n=-3000}^{3000} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2}(n \times 0.001)^2\right] \times 0.001 \] **Explanation:** For each value of \(\Delta\), we are partitioning the interval \([-3, 3]\) into smaller and smaller steps, making the approximation increasingly accurate as \(\Delta\) decreases. --- ## **Final Answer** The value of the integral \[ \int_{-3}^{3} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2} t^2\right] dt \] can be approximated by the sum \[ \sum_{n=-L}^{L} \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{1}{2}(n\Delta)^2\right] \Delta \] where \(L\) is the integer closest to \(3/\Delta\). **For each \(\Delta\):** - For \(\Delta = 0.1\), \(L = 30\) - For \(\Delta = 0.01\), \(L = 300\) - For \(\Delta = 0.001\), \(L = 3000\) Plug in your desired \(\Delta\) and compute the sum for an increasingly accurate approximation!