In Exercises 31 – 32, approximate the value of the given definite integral by using the first ϰ nonzero terms of the integrand’s Taylor series.
\[ A = \int_{0}^{\sqrt{x}}sin(x^2)dx \]
\[ A = \int_{0}^{\frac{x^2}{4}}cos(\sqrt{x})dx \]
To approximate the value of the given definite integrals using the first κ nonzero terms of the integrand’s Taylor series,
we will find the Taylor series expansions of the integrands and then integrate each term separately.
\[ A = \int_{0}^{\sqrt{x}}sin(x^2)dx \]
The Taylor series expansion of sin(x^2) around x=0 is:
\[sin(x^2) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!}\] Now, integrate each term of the series separately:
\[A \approx \int_{0}^{\sqrt{x}}(x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!}-...)dx\]
Integrate term by term and sum them up to get the final approximation.
\[ A = \int_{0}^{\frac{x^2}{4}}cos(\sqrt{x})dx \]
The Taylor series expansion of cos(sqrt(x)) around x=0 is:
\[cos (\sqrt{x})= 1-\frac{x}{2} + \frac{x^2}{4!}-...\] Next, we integrate each term of the series separately:
\[A \approx \int_{0}^\frac{x^2}{4}(1-\frac{x}{2} + \frac{x^2}{4!}-...)dx\]
Integrate term by term and sum them up to get the final approximation.