Question

Evaluate ∫f 𝑜𝑣𝑒𝑟 𝑐 where 𝑓( 𝑧 )= 𝑥^2 + 𝑖𝑦^2 where c is given by 𝑧 (𝑡 )= 𝑡^2 + 𝑖𝑡^2, 0 ≤ 𝑡 ≤ 1

Answer 1

$\int_C{f}=\left[ \begin{array}{c} z\left( t \right) =t^2+it^2\\ dz\left( t \right) =\left( 2t+2it \right) dt=2\left( 1+i \right) tdt\\ f\left( z \right) =x^2+iy^2=\\ =\left( t^2 \right) ^2+i\left( t^2 \right) ^2=\\ =\left( 1+i \right) t^4\\\end{array} \right] =\int_0^1{\left( 1+i \right) t^4\cdot 2\left( 1+i \right) tdt}=\\=2\left( 1+i \right) ^2\int_0^1{t^5dt}=\frac{2\cdot 2i}{6}=\frac{2}{3}i$