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

Question:

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

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.

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{eq}{/eq}

The definite integral is given as, {eq}\int_{0}^{1}\sqrt{1 + x^4}dx &\text{[1]}\\ {/eq}. To approximate the result of this definite...

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Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.


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