Use series to approximate {eq}\int_{0}^{1}\sqrt{1 + x^4}dx {/eq} correct to two decimal places.

Question:

Series Approximation of an Integral

{eq}{/eq}

• A definite integral of a function {eq}f(x) {/eq} can be estimated by the use of series expansion. Given the condition that the function {eq}f(x) {/eq} has a series representation.

• As an example, if the function has the form, $$(a+b)^n $$then the function can be expressed into a binomial series given as,

$$\begin{align*} &(a+b)^n=\sum_{i=0}^{n}\binom{n}{i}a^{n-i}b^i\\\\ &\binom{n}{i}=\frac{n(n-1)(n-2)\cdots (n-i+1)}{i!} \end{align*} $$

• Once the function has been represented into a series, the limits of integration can then be substituted into the series, thereby, approximating the definite integral operation.

Answer and Explanation: 1