Author Topic: Quiz3 problem 6E  (Read 269 times)

Xuefen luo

  • Full Member
  • ***
  • Posts: 18
  • Karma: 0
    • View Profile
Quiz3 problem 6E
« on: October 18, 2020, 03:52:58 PM »
Problem: Compute the following line integral:
\begin{align*}\int_{\gamma}^{} |z|^2 \,dz, \end{align*}
where $\gamma$ is the line segment from 2 to 3 + i
\begin{align*}
\end{align*}
Answer:
\begin{align*}
 \gamma(t)&= (1-t)z_0+tz_1\\
 &= 2(1-t)+(3+i)t\\
 &= 2-2t+3t+it\\
 &= 2+t+it \ \ \ \  (0 \leq t \leq 1)\\
 \\
 \gamma'(t)&=1+i\\
 \\
 Let\ f(z)=|z|^2\\
 \\
 f(\gamma(t))&= |\gamma(t)|^2\\
 &=(\sqrt{(2+t)^2+t^2})^2\\
 &=4+4t+t^2+t^2\\
 &=2t^2+4t+4\\\
 \\
 \int_{\gamma}^{} |z|^2 \,dz &= \int_{\gamma}^{} f(z) \,dz\\
 &=\int_{0}^{1} f(\gamma(t))\  \gamma'(t) \,dt\\
 &=\int_{0}^{1} (2t^2+4t+4) (1+i) \,dt\\
  &=(1+i)\int_{0}^{1} (2t^2+4t+4) \,dt \\
   &=(1+i)(\frac{2}{3}t^3 + 2t^2 +4t)|_{0}^{1}\\
    &=(1+i)(\frac{2}{3} + 2 +4)\\
     &=\frac{20}{3}(1+i)
\end{align*}