Solve the following using the parametric equations

Question

Consider the parametric curve over the interval [0, pi/3]:

𝑥 = 4cos^2(2𝑡)

𝑦 = cot^2(3𝑡) + 1

a. Set up and simplify an integral describing the area under the curve from (0, 2) to (3, 2)

b. Set up and simplify the integral describing the length of the curve from (0, 2) to (1, 1).

Answer 1

Answer 2

A = ∫ y(t) x'(t) d t from t₁= π/4 to t₂ = π/12

∫ _t1^t2 [cos²(3t) +1][ - 8 sin(4t) ] d t

∫ _t2^t1 [cos²(3t) +1][ 8 sin(4t) ] d t

3[ 2√3 −15] / 10

S = ∫₀^π/4 {[ x'(t)]² + [y'(t)]² }^1/2 d t =

∫₀^π/4 {[ 8 sin (4t) ]² + [3 sin(6t)]² }^1/2 d t