1.7 Show that the circumference of the ellipse x2/a2 + y2/b2 = 1 is 5 = 40E - (a2 -63)1/2 where E(k, ) =fs (1 -*3 sin )" denotes the standard "elliptic integral of the second kind."Certainly! The circumference of an ellipse cannot be expressed in terms of elementary functions, which is why we use elliptic integrals to represent it. The equation you provided involves an elliptic integral of the second kind, denoted E(k, 4). Here's a step-by-step explanation of the terms and the equation provided: Ellipse Equation: The standard form of an ellipse is * + # - 1, where a and b are the lengths of the semi-major and semi-minor axes, respectively. 2. Circumference of the Ellipse: The formula for the circumference of an ellipse is given by the integral: s = 4aE(e, T/2) Here, a is the semi-major axis, and _ is the complete elliptic integral of the second kind. Elliptic Integral of the Second Kind: The elliptic integral of the second kind, E (k, ), is defined as: E(k, 4) V1 - k2 sin- 0 de where is the amplitude (in this case, 7 /2 for the complete elliptic integral), and k is the elliptic modulus or eccentricity of the ellipse, defined as k* = 1 - ". Complete Elliptic Integral: When d - 7 /2, E(k, () becomes E (k), the complete elliptic integral of the second kind, which is dependent only on k.5. Eccentricity (k): The eccentricity k. of an ellipse is a measure of how much the shape deviates \\ from being circular. For an ellipse if; -l lg; l. the square of the eccentricity is given by k2 1 3]. The equation you've given calculates the circumference 5 using the complete elliptic integral of the second Kind. It takes into account the non-circular shape of the ellipse to accurately measure the perimeter. This integral does not have a simple antiderivative and istypically evaluated numerically or using series expansions