{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² sin x² dx

Verified step by step guidance

1. Recognize that the integral involves the function \( \sin(x^2) \), which does not have an elementary antiderivative, making a Taylor series approximation a suitable approach.
2. Write the Taylor series expansion of \( \sin z \) around \( z=0 \): \[ \sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots \] Substitute \( z = x^2 \) to get the series for \( \sin(x^2) \): \[ \sin(x^2) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \cdots \]
3. Set up the integral of the series term-by-term from 0 to 0.2: \[ \int_0^{0.2} \sin(x^2) \, dx = \int_0^{0.2} \left( x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \cdots \right) dx = \sum_{n=0}^{\infty} (-1)^n \frac{1}{(2n+1)!} \int_0^{0.2} x^{4n+2} \, dx \]
4. Integrate each term using the power rule: \[ \int_0^{0.2} x^{m} \, dx = \frac{(0.2)^{m+1}}{m+1} \] where \( m = 4n + 2 \). So each term becomes: \[ (-1)^n \frac{(0.2)^{4n+3}}{(2n+1)! (4n+3)} \]
5. Add terms of the series until the absolute value of the next term is less than \( 10^{-4} \) to ensure the error is within the required tolerance. Sum all included terms to approximate the value of the integral.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For sin(x²), expanding around x=0 allows approximation by polynomials, making integration manageable. Truncating the series after enough terms controls the approximation error.

Definite Integral Approximation Using Series

Integrating a function approximated by a Taylor series involves integrating each polynomial term individually over the given interval. This method simplifies complex integrals into sums of integrals of powers of x, which are straightforward to compute, enabling approximate evaluation of the definite integral.

Error Estimation and Control

When truncating a Taylor series, the remainder term estimates the error between the true function and its polynomial approximation. Ensuring the error is less than 10⁻⁴ requires calculating or bounding this remainder, guiding how many terms to retain for a sufficiently accurate integral approximation.