Arc lengths leading to elliptic integrals

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Problem 2453 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 4.6 "Computation of Arc Length"

Prove that the arc length of the ellipse

x=a\cos t ,\quad y=b\sin t

equals the length of one wave of the sine curve $y=c\sin \dfrac {x}{b}$ , where $c=\sqrt{a^{2}-b^{2}}$ .

Problem 2544 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 4.11 "Approximate Methods for Evaluating Definite Integrals"

Approximately find the perimeter of an ellipse with semi-axes $a=10$ and $b=6$ .

Problem 2933 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.5 "Power Series"

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

Problem 3043 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.8 "Evaluating Definite Integrals Using Series"

Express the arc length of the following ellipse as a series expanded in positive integer powers of the eccentricity:

x=a\cos t,\quad x=b\sin t \quad (0 \leqslant t \leqslant 2\pi ).

Problem 2.7.98 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.7.2 "Problems"

Express, in terms of elliptic integrals, the length of the arc of the curve:

(1)

$y = a \sin \omega x, 0 \leqslant x \leqslant x_{0} \leqslant \pi /(2 \omega )$ ;

(2)

$x = a t-b \sin t, y = a-b \cos t, a > 0, b > 0,0 \leqslant t_{0} \leqslant t \leqslant \pi$ ;

(3)

$r^{2} = 2 a^{2} \cos 2 \varphi , 0 \leqslant \varphi \leqslant \varphi_{0} < \pi / 4$ (lemniscate);

(4)

$x^{2} / a^{2}+y^{2} / b^{2} = 1,0 \leqslant x \leqslant x_{0} \leqslant a, y \geqslant 0, a > b$ ;

(5)

$x^{2} / a^{2}-y^{2} / b^{2} = 1,0 \leqslant y \leqslant y_{0}, x > 0$ .

Problem 2.7.99 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.7.2 "Problems"

(1)(2)(3)(4)

Find the length of the curve:

(1)

$x^{2} / a^{2}+y^{2} / b^{2} = 1$ ;

(2)

$r^{2} = 2 a^{2} \cos 2 \varphi$ ;

(3)

$r = a \cos \varphi +b$ (limaçon);

(4)

$r = a \sin n \varphi , \quad n \in \mathbb {N}, \quad n > 1$ .

Problem 2.7.100 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.7.2 "Problems"

Prove that the length of the ellipse $x^{2} / a^{2}+y^{2} / b^{2} = 1, a > b$ , equals the length of the sinusoid $y = \sqrt{a^{2}-b^{2}} \sin (x / b), 0 \leqslant x \leqslant 2 \pi b$ .

Problem 2.7.101 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.7.2 "Problems"

Prove that the length of the arc of the curve

$x = a t-b \sin t, \quad y = a-b \cos t, \quad 0 \leqslant t \leqslant 2 \pi , \quad a > 0, \quad b > 0$ , equals the length of the ellipse with semi-axes $a+b$ and $\left|a-b\right|$ .

Problem 2.7.102 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.7.2 "Problems"

Prove that the length $s$ of an ellipse with semi-axes $a$ and $b$ satisfies the inequalities

\pi (a+b) < s < \pi \sqrt{2\left(a^{2}+b^{2}\right)}

Problem 2.7.111 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03a]More problems in Subsection 2.7.2 "Problems"

Find the length of Viviani's curve

x = R \sin ^{2} t, \quad y = R \sin t \cos t, \quad z = R \cos t.

Problem 3.10.84 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03b]More problems in Subsection 3.10.2 "Problems"

Using tables, find, with an error no greater than 0.1, the arc length of the curve $x^{2}+y^{2}+z^{2} = 1, y^{2}+z^{2} = y$ .

Problem 4.124.1 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 4.124 "[NO TITLE]"

Let $l$ denote the length of the perimeter of an ellipse with unequal semi-axes $a$ and $b$ . Give two different proofs (with and without the use of conformal mapping) for the inequality

l > \pi (a+b).

Remark.

The right-hand side is a convenient approximation to $l$ when $a$ and $b$ are nearly equal.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ККЧ+03a] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.

[ККЧ+03b] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.