Approximate computation via series

Evaluate the integral

to an accuracy of two decimal places; that is, find a number $I^{*}$ such that $\left|I-I^{*}\right| < 0.005$.

Estimate the absolute error of the following approximate formulas:

For what values of $x$ is the approximate formula

accurate to $0.0001$.

Using Taylor's formula, compute approximately, and estimate the error:

Compute:

Compute

accurate to $10^{-4}$.

For which $n$ can the inequality

be guaranteed to hold?

Using Newton's binomial formula, approximately compute $\sqrt[3]{9}$, and estimate the error when only the first three terms of the expansion are taken.

Compute approximately, and estimate the corresponding error:

Using a suitable expansion, compute $\sin 18°$ accurate to $10^{-5}$.

Using a suitable expansion, compute $\cos 1°$ accurate to $10^{-6}$.

Using a suitable expansion, compute $\tan 9°$ accurate to $10^{-3}$.

Using a suitable expansion, compute $e$ accurate to $10^{-6}$.

Using a suitable expansion, compute $\ln 1.2$ accurate to $10^{-4}$.

From the equality $\dfrac {\pi }{6}=\arcsin \dfrac {1}{2}$, find the number $\pi$, accurate to $10^{-4}$.

Using the identity $\dfrac {\pi }{4}=\arctan \dfrac {1}{2}+\arctan \dfrac {1}{3}$, compute the number $\pi$, accurate to $0.001$.

Using the formula $\dfrac {\pi }{4}=4\arctan \dfrac {1}{5}-\arctan \dfrac {1}{239}$, find the number $\pi$, accurate to $10^{-9}$.

Using the formula $\ln (n+1)=\ln n+2\left(\dfrac {1}{2n+1}+\dfrac {1}{3(2n+1)^{3}}+\cdots \right)$, find $\ln 2$ and $\ln 3$, accurate to $10^{-5}$.

Use the series expansion of the integrand to compute the value of each of the following integrals, accurate to $0.001$:

Find the arc length of a single half-wave of the sine curve $y=\sin x$ ($0 \leq x \leq \pi$), accurate to $0.01$.

An ellipse has semi-axes $a=1$ and $b=\dfrac {1}{2}$; find the arc length of the ellipse, accurate to $0.01$.

A wire is strung between two poles standing a distance $2l=20$ meters apart, and the wire hangs in the shape of a parabola. Given that the sag at the lowest point is $h=40$ centimeters, compute the length of the wire, accurate to $1$ centimeter.

Using Taylor's formula with the remainder term in Lagrange's form (25), approximately compute (to an accuracy of $10^{-3}$):

Using Taylor's formula (25), estimate the absolute error of the approximate formulas:

Using Taylor's formula (25), compute:

Compute the integral $\int_{0}^{1} \sin \frac{x^{2}}{2} d x$ with error less than $10^{-5}$.

Compute the integral $\int_{0}^{1} \cos x^{2} d x$ to five correct digits

Prove that:

Prove that $0 < \int_{0}^{1} e^{-x^{2}} d x-\frac{26}{35} < 0,005$.

Prove that $\int_{\pi / 4}^{\pi / 2} \frac{\sin x}{x} d x \approx 0,610$ with error not exceeding $1,5 \cdot 10^{-3}$.

Prove that for $0 < z < 1$, for the integral $J = \int_{0}^{z} \frac{\sqrt[3]{x} d x}{1+x^{4}}$ the inequalities $\frac{3}{4} z^{4 / 3}-\frac{3}{16} z^{16 / 3} < J < \frac{3}{4} z^{4 / 3}-\frac{3}{16} z^{16 / 3}+\frac{3}{28} z^{28 / 3}$ hold.

Compute the integral $\int_{0}^{1} \frac{\sin x}{x} d x$ with error less than $10^{-5}$, using Taylor's formula.

Compute the integral $\int_{0}^{1 / 4} \frac{\ln (1+x)}{x} dx$ with error less than $10^{-4}$, using Taylor's formula.

Compute the integral $\int_{-1}^{1} \sqrt{8+x^{3}} d x$ with error less than $10^{-4}$, using Taylor's formula.

Compute the length of a quarter of the ellipse $x^{2}+2 y^{2} = 2$, using Taylor's formula with three terms.

Compute the integral $\int_{0}^{1} \sqrt[4]{1-x^{2}} d x$ with error not exceeding $10^{-3}$.

Compute the integral $\int_{0}^{1} \frac{\sin x}{1+x^{2}} d x$ with error less than $10^{-3}$.

Prove that for any $n \in \mathbb {N}, n \geqslant 2$,

where $a_{n} = 1+\frac{1}{n(n+1)}$.

Compute the integral $\int_{0}^{1} \sqrt[10]{1+x^{10}} d x$ with error not exceeding $10^{-3}$.

Prove that $n\left(a_{k}-\frac{3(k-1)}{32 k^{2}}\right) < \int_{0}^{n} \sqrt[k]{1+\sin^{2} \pi x} d x < n a_{k}$, where $a_{k} = 1+1 /(4 k), n \in \mathbb {N}, k \in \mathbb {N}, k \geqslant 2$.

Prove that $1,0473 < \int_{0}^{1} \sqrt[3]{1+\sin^{4} \pi x} d x < 1,0625$.

Compute the integral $\int_{0}^{8} \sqrt{1+\sin^{6} \pi x} dx$ with error not less than $10^{-3}$.

Using the corresponding expansions, compute to within $10^{-4}$:

Compute to within $10^{-3}$:

Compute to within $10^{-5}$:

Using the equality $\pi / 6 = \arcsin (1 / 2)$, compute the number $\pi$ by means of the Maclaurin series for the function $\arcsin x$ to within $10^{-4}$.

Using the equality

compute the number $\pi$ to within $10^{-9}$.

Using the Maclaurin series for $f(x)$, compute the integral $\int_{0}^{1} f(x) d x$ to within $10^{-3}$, if:

Compute to within $10^{-3}$: