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E2-4 مرحلة ثانية شعبة رانية إشعار واحد
Answer two questions only.
Q1: Solve the equation
\[
z^{4}=-1+i
\]
Q2: let \( z=x y+\sin x \). Find the total derivative.
Q3: Evaluate the double integral
\[
\iint_{R}\left(2 \frac{x^{2}}{y^{2}}+2 y\right) d A
\]
Where \( R: \begin{array}{l}1 \leq x \leq 2 \\ 1 \leq y \leq x\end{array} \)

Answer

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Answer

\[R: \begin{array}{l}1 \leq x \leq 2 \\ 1 \leq y \leq x\end{array}\]

Steps

Step 1 ：Q1:

Step 2 ：\[z^4=-1+i\]

Step 3 ：Q2:

Step 4 ：\[z = xy + \sin x\]

Step 5 ：\[\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}\]

Step 6 ：Q3:

Step 7 ：\[\iint_{R}\left(2 \frac{x^{2}}{y^{2}}+2 y\right) d A\]

Step 8 ：\[R: \begin{array}{l}1 \leq x \leq 2 \\ 1 \leq y \leq x\end{array}\]