Evaluate the following line integral over the curve given by x = t2 , y = t3 , where 1 ≤t ≤ 2.∫ ! !""#𝑑𝑠

Find the line integral of 𝑓(𝑥, 𝑦) = 𝑦𝑒 # !along the curve r(t) = 4t i – 3t j, -1 ≤ t ≤ 2

The line integral ∫CF⃗ ⋅dr⃗ ∫𝐶𝐹→⋅𝑑𝑟→, where F⃗ =xyi⃗ +yzj⃗ +zxk⃗ 𝐹→=𝑥𝑦𝑖→+𝑦𝑧𝑗→+𝑧𝑥𝑘→ and C𝐶 is the curve given by r⃗ (t)=ti⃗ +t2j⃗ +t3k⃗ 𝑟→(𝑡)=𝑡𝑖→+𝑡2𝑗→+𝑡3𝑘→, 0≤t≤10≤𝑡≤1 is expressed as the definite integral

Evaluate the line integrals ∫𝐶 𝑓 (𝑧)𝑑𝑧 along the curve 𝐶, where(a) 𝑓 (𝑧) = 1𝑧 − 2 , 𝐶 : |𝑧| = 3 in the counterclockwise direction.(b) 𝑓 (𝑧) = 𝑧3 , 𝐶 : |𝑧 − 𝑖| = 2 in the counterclockwise direction.(c) 𝑓 (𝑧) = 1𝑧2 + 1 , 𝐶 : |𝑧| = 1 in the counterclockwise direction.(d) 𝑓 (𝑧) = |𝑧|2 , 𝐶 : 𝑡2 + 𝑖𝑡 where 0 ≤ 𝑡 ≤ 1.

To evaluate the given double integral over the region 𝐷D, bounded by the lines 𝑦=−𝑥y=−x, 𝑦=𝑥2y=x 2 , and 𝑦=2y=2,

The line integral over C, the arc of the parabola x = y2 from (1,-1) to (1,1)