Using Simpson's Rule with n = 4, the approximated value of $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x,$ to three decimal places, is

A) $0.747$
B) $0.176$
C) $0.285$
D) $0.359$
E) $0.237$

Question

Using Simpson's Rule with n = 4, the approximated value of $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x,$ to three decimal places, is

A) $0.747$
B) $0.176$
C) $0.285$
D) $0.359$
E) $0.237$

Quizplus · Accepted Answer

Simpson's Rule is a method for numerical integration that approximates the area under a curve. For $n = 4$ (even number of intervals), the formula is $\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 2\sum_{i=1}^{n/2-1}f(x_{2i}) + 4\sum_{i=1}^{n/2}f(x_{2i-1}) + f(x_n)]$, where $\Delta x = \frac{b-a}{n}$. Applying this to $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x$ with $n = 4$, the calculation yields an approximation close to $0.747$, making choice A correct.

Using Simpson's Rule with N = 4, the Approximated Value ∫01e−x2dx,\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x,∫01​e−x2dx,

Using Simpson's Rule with N = 4, the Approximated Value $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x,$