Using the Trapezoidal Rule with n = 3, the approximated value of $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x,$ to three decimal places, is

A)1.414
B)1.248
C)1.093
D)0.983
E)0.895

Question

Using the Trapezoidal Rule with n = 3, the approximated value of $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x,$ to three decimal places, is

A)1.414
B)1.248
C)1.093
D)0.983
E)0.895

Quizplus · Accepted Answer

The Trapezoidal Rule with n = 3 is given by:$$\int_a^b f(x) dx \approx \frac{b-a}{2n} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$$where $x_i = a + ih$ and $h = \frac{b-a}{n}$.In this case, $a = 0$, $b = 1$, $n = 3$, and $f(x) = e^{x^2}$. So, we have:$$h = \frac{1-0}{3} = \frac{1}{3}$$$$x_0 = 0$$$$x_1 = \frac{1}{3}$$$$x_2 = \frac{2}{3}$$$$x_3 = 1$$$$\int_0^1 e^{x^2} dx \approx \frac{1}{6} [e^0 + 2e^{(\frac{1}{3})^2} + 2e^{(\frac{2}{3})^2} + e^1]$$$$\approx \frac{1}{6} [1 + 2e^{\frac{1}{9}} + 2e^{\frac{4}{9}} + e^1]$$$$\approx 1.414$$Therefore, the approximated value of $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x,$ to three decimal places, is 1.414.

Using the Trapezoidal Rule with N = 3, the Approximated ∫01ex2dx,\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x,∫01​ex2dx,

Using the Trapezoidal Rule with N = 3, the Approximated $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x,$