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Chapter 1: Problem 18

Evaluate the integral \(\int_{0}^{\pi / 6} \sin ^{2} x d x\) by doing the following: a. Compute the integral exactly. b. Integrate the first three terms of the Maclaurin series expansion of the integrand and compare with the exact result.

Short Answer

Expert verified
The exact integral can be calculated using trigonometric identities while the approximate integral is found by integrating the first three terms of the Maclaurin series expansion. A comparison of these results will reveal the accuracy of the Maclaurin series approximation.

Step by step solution

01

Compute the integral exactly

The integral \(\int_{0}^{\pi / 6} \sin^{2}(x) dx\) involves a trigonometric function. First, it's important to express \(\sin^{2}(x)\) in terms of cosine using the identity \(\sin^{2}(x)=\frac{1-\cos (2 x)}{2}\). Then, we integrate from 0 to \(\pi/6\) to obtain the exact result.
02

Find the Maclaurin series expansion

The Maclaurin series expansion for \(\sin(x)\) is \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\). The square of this sequence gives the expansion for \(\sin^{2}(x)\). Collect the first three terms of this expansion.
03

Integrate the first three terms of the Maclaurin series

Next, the first three terms of the Maclaurin series expansion of \(\sin^{2}(x)\) will be integrated from 0 to \(\pi/6\).
04

Compare results

Once the integrals are calculated, both results should be compared to see how accurate the Maclaurin series approximation is in relation to the exact result.

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Most popular questions from this chapter

Consider Gregory's expansion $$ \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\cdots=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2 k+1} x^{2 k+1} $$ a. Derive Gregory's expansion using the definition $$ \tan ^{-1} x=\int_{0}^{x} \frac{d t}{1+t^{2}} $$ expanding the integrand in a Maclaurin series, and integrating the resulting series term by term. b. From this result, derive Gregory's series for \(\pi\) by inserting an appropriate value for \(x\) in the series expansion for \(\tan ^{-1} x\).

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Evaluate the following expressions at the given point. Use your calculator or your computer (such as Maple). Then use series expansions to find an approximation of the value of the expression to as many places as you trust. a. \(\frac{1}{\sqrt{1+x^{3}}}-\cos x^{2}\) at \(x=0.015\). b. \(\ln \sqrt{\frac{1+x}{1-x}}-\tan x\) at \(x=0.0015\). c. \(f(x)=\frac{1}{\sqrt{1+2 x^{2}}}-1+x^{2}\) at \(x=5.00 \times 10^{-3}\). d. \(f(R, h)=R-\sqrt{R^{2}+h^{2}}\) for \(R=1.374 \times 10^{3} \mathrm{~km}\) and \(h=1.00 \mathrm{~m}\). e. \(f(x)=1-\frac{1}{\sqrt{1-x}}\) for \(x=2.5 \times 10^{-13}\).
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