Show that the total length of the ellipse x = a sin(8), y = b cos(8), a > b > 0, is TT/2 ™¹²2 √ ₁ – e² sin²(0) dº L = 4a where e is the eccentricity of the ellipse described by e = y (e=, where c here c = √3² - 6²). a

Q: Given the parametric equations: I = e' - t, y = 4e/², with 0 < t< 2 (a.) Find dy and dy as functions…

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Q: Show that the curve y = x? sin 3x touches the parabolas y = x andy =-x² alternately at infinite…

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Q: Find F. dr for given F and C. (a) (2,4). F = −y sin(x)i + cos(x)] and C is the parabola y = x²…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: dy Write an equation for dx on the ellipse y = y² + x2 3, showing ALL - your work. Then find the…

A: We differentiate both sides with respect to x and then solve for dy/dx.

Q: A curve is given parametrically by the equations x = 3t - 2 sint and y = 1² + t cos t Find the exact…

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Q: Find the length of the hypocycloid defined by the equations x = a cos° (0), y = a sin° (0), 0 < 0 <…

A: Here we have to find the length of the hypocycloid defined by the equation x=acos3(θ) , y=asin3(θ)

Q: Parameterize the curve r(t) = (2+ cos 3t, 3 – sin 3t, 4t) using arc length, s, as a parameter. Use t…

A: Given the curve rt=2+cos3t, 3-sin3t, 4t---1. Differentiating, we get vt=r't=-3sin3t,…

Q: The path of a projectile is modeled by the parametric equations x = (90cos(60°)) and y= (90sin(60°))…

A: The path of projectile described by the curveTo find the arc length of the projectile.We use the…

Q: 4. Let P be any point on the ellipse E: x²y² + = 1. 25 9 (i) Show that the equation of the tangent…

A: Two straight line is parallel to each other if their slopes are equal.

Q: 2. Write the equation (do simplify, but do not integrate!) for the arclength of the graph x=e¹y=e²¹…

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Q: A curve C is given parametrically by x = cos(t) - 2 and y = sin(t) + 3, where 0 ≤t≤ 2π Find and…

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Q: Find the length of the curve defined by the parametric equations t, y = 5 In((t/6)? – 1) I = from t…

A: Given, x = 56t , y = 5lnt62-1 = 5lnt2-36-5ln(36)and 7≤t≤10

Q: - Find the area of the ellipse which is parametrically expressed as =a cos o and y=b sin ø, and…

A: To find the area of the ellipse which is parametrically expressed by the below equations.…

Q: Find the arclength L(C) of the helix C parametrized by C: r(t) = (2 cos t, -3t, 2 sin t) for 0 <t <…

A: The arc length of the parametric curve is given by: ∫ab(x'(t))2+(y'(t))2+(z'(t))2dt...(1); a≤t≤b…

Q: Q2: Gruph the nector fundion Tt)= Find the equat in for the tanget f functron fexy) = SinCxy) at…

A: a) Given,  rt=cost, sint, t Here  xt=cost,yt=sint,zt=t