Calculus Answers

Evaluate the line integral ∫𝒖(𝑥, 𝑦, 𝑧) × ⅆ𝒓 𝐶 , where 𝒖(𝑥, 𝑦, 𝑧) = (𝑦 2 , 𝑥, 𝑧) and the curve 𝑪 is described by 𝒛 = 𝑦 = 𝑒 𝑥 with 𝑥 ∈ [0,1].

Find the curvature, the radius and the center of curvature at a point.

r=1+ cos theta ,theta=π/2

Find the exact arc length of the curve

x = e ^ 2t* (sin t + cos t) , y=e^2t(sin t - cos t) ; (−1≤t≤1)

Find the exact arc length of the curve

x = cos 3t, y = sin 3t ;(0 ≤ t ≤π)

Find the equation of Tangent line to the curve ;

x=e^t , y=e^-t at t=1

Find dy/dx and d²y/dx² of;

x = theta + cos theta, y = 1 + sin theta

; theta = pi/6

Find dy/dx and d²y/dx²;

x=√t, y=2t+4 ; t=1

Find the curvature, the radius of curvature and the center of curvature having a parametric equations at the given point:

x = sin y, (1/2,1/6 π)

Find the slope of the curve and the equation of tangent line of the parametric equation to the given point.

x=ln t , y = t ^ - 1 , when t = 2