yConsider the ellipse1.(a) Show that the circumference of the ellipse is given by C4 72 V4 cos? t+9 sin? t dt.CT/2(b) Show that this integral can be simplified to C8 2 V1+ sin? t dt.%3D1+5u²/4(c) By taking u=sin(t), show that it further simplifies to C = 8du. (Be careful, as this is an improper1-и2integral.)x dx(d) Now make a substitution x = ? to find C = 8 /4(Also improper.)Vx(x-1)(9-4x)We can think of the denominator here as related to the elliptic curve y? = x(x – 1)(9 – 4x), and in the sense ofdifferential geometry we are really integrating a differential form on this elliptic curve over a loop. Integrals such asthese are called elliptic integrals, and they cannot be evaluated in terms of elementary functions. But this is howelliptic curves received their name.

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Consider the ellipse + = 1. n/2 Show that the circumference of the ellipse is given by C 2 V4 cos? t + 9 sin? t dt. = 4 /2 Show that this integral can be simplified to C = 8 S" V1+sin? t dt. 1+5u²/4 By taking u=sin(t), show that it further simplifies to C = 8 du. (Be careful, as this is an improper 1-u? cegral.) c9/4 Ji æ dx Now make a substitution x = ? to find C = 8 4 (Also improper.) Vr(x-1)(9-4x) e can think of the denominator here as related to the elliptic curve y? = x(x – 1)(9 – 4x), and in the sense of ferential geometry we are really integrating a differential form on this elliptic curve over a loop. Integrals such as ese are called elliptic integrals, and they cannot be evaluated in terms of elementary functions. But this is how iptic curves received their name.

Consider the ellipse + = 1. n/2 Show that the circumference of the ellipse is given by C 2 V4 cos? t + 9 sin? t dt. = 4 /2 Show that this integral can be simplified to C = 8 S" V1+sin? t dt. 1+5u²/4 By taking u=sin(t), show that it further simplifies to C = 8 du. (Be careful, as this is an improper 1-u? cegral.) c9/4 Ji æ dx Now make a substitution x = ? to find C = 8 4 (Also improper.) Vr(x-1)(9-4x) e can think of the denominator here as related to the elliptic curve y? = x(x – 1)(9 – 4x), and in the sense of ferential geometry we are really integrating a differential form on this elliptic curve over a loop. Integrals such as ese are called elliptic integrals, and they cannot be evaluated in terms of elementary functions. But this is how iptic curves received their name.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.1: Parabolas

Problem 65E

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