Evaluate the following line integral over the curve given by x = t2 , y = t3 , where 1 ≤t ≤ 2

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 integral  CF · dr, where C is given by the vector function r(t).F(x, y, z) = (x + y2) i + xz j + (y + z) k,r(t) = t2 i + t3 j − 2t k,  0 ≤ t ≤ 2

The area enclosed by the lines x = 0, x = 2, y = x and y = 3 is

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.