Chapter 8: Problem 1
If the interval [4,18] is partitioned into \(n=28\) sub-intervals of equal length, what is \(\Delta x ?\)
Over 30 million students worldwide already upgrade their learning with Vaia!
These are the key concepts you need to understand to accurately answer the question.
All the tools & learning materials you need for study success - in one app.
Get started for free
It can be shown that $$\begin{array}{l}\int_{0}^{\pi / 2} \sin ^{n} x d x=\int_{0}^{\pi / 2} \cos ^{n} x d x= \\\\\quad\left\\{\begin{array}{ll}\frac{1 \cdot 3 \cdot 5 \cdot \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2} & \text { if } n \geq 2 \text { is an eveninteger } \\\\\frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdots n} & \text { if } n \geq 3 \text { is an odd integer. }\end{array}\right.\end{array}$$ a. Use a computer algebra system to confirm this result for \(n=2,3,4,\) and 5 b. Evaluate the integrals with \(n=10\) and confirm the result. c. Using graphing and/or symbolic computation, determine whether the values of the integrals increase or decrease as \(n\) increases.
Period of a pendulum A standard pendulum of length \(L\) that swings under the influence of gravity alone (no resistance) has a period of $$ T=\frac{4}{\omega} \int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-k^{2} \sin ^{2} \varphi}} $$ where \(\omega^{2}=g / L, k^{2}=\sin ^{2}\left(\theta_{0} / 2\right), g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(\theta_{0}\) is the initial angle from which the pendulum is released (in radians). Use numerical integration to approximate the period of a pendulum with \(L=1 \mathrm{m}\) that is released from an angle of \(\theta_{0}=\pi / 4\) rad.
Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of \(n .\) a. Find the Trapezoid Rule approximations to the integral using \(n\) and \(2 n\) subintervals. b. Find the Simpson's Rule approximation to the integral using \(2 n\) subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(8 .\) c. Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with \(2 n\) subintervals. $$\int_{1}^{e} \frac{d x}{x} ; n=50$$
For a real number \(a\), suppose \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\) or \(\lim _{x \rightarrow a^{+}} f(x)=\infty .\) In these cases, the integral \(\int_{a}^{\infty} f(x) d x\) is improper for two reasons: \(\infty\) appears in the upper limit and \(f\) is unbounded at \(x=a .\) It can be shown that \(\int_{a}^{\infty} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{\infty} f(x) d x\) for any \(c>a .\) Use this result to evaluate the following improper integrals. $$\int_{1}^{\infty} \frac{d x}{x \sqrt{x-1}}$$
Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=\cos x^{2}\) a. Find a Midpoint Rule approximation to \(\int_{-1}^{1} \cos x^{2} d x\) using \(n=30\) subintervals. b. Calculate \(f^{\prime \prime}(x)\) c. Explain why \(\left|f^{*}(x)\right| \leq 6\) on [-1,1] d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
