Use series to approximate the definite integral to within the indicated accuracy. $$\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid

Question

Use series to approximate the definite integral to within the indicated accuracy. $$\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$$
A) $0.0354$
B) $0.0125$
C) $0.0625$
D) $0.1447$
E) $0.2774$

Quizplus · Accepted Answer

The function $e^{-x^2}$ can be expanded as a power series: $e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \frac{x^6}{3!} + \cdots$. Integrating term by term from 0 to 0.5 and ensuring the error is less than 0.001, the approximate value of the integral is 0.0354.

Use Series to Approximate the Definite Integral to Within the Indicated