2. 2.1. กำหนดให้ \( f(x, y)=\frac{\ln \left(4-x^{2}-y^{2}\right)}{\sqrt{y^{2}-x+1}} \) จงวาดบริเวณโดเมนของฟังก์ชัน \( f \) 2.2. จงวาดกราฟอย่างคร่าวๆ ของพื้นผิว \( y^{2}=\frac{x^{2}}{9}+\frac{z^{2}}{4} \) 2.3. กำหนดให้ \( g(x, y)=1-x^{2}+y^{2} \) จงวาดเส้นโค้งระดับ \( g(x, y)=k \) เมื่อ \( k=0,1,2 \)

\( \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } e ^ { \frac { x } { y } } d y d x \)

put \( z=\mathrm{Re}^{i \theta} \), then \( |z|=R \) Now, \( \left|\int_{\Gamma} f(z) d z\right| \leq \frac{\pi R}{\left(R^{2}-1\right)^{2}} \rightarrow 0 \) as \( R \rightarrow \infty \) Thus, the equation (1) becomes \( \lim _{R \rightarrow \infty} \int_{-R}^{R} f(x) d x+\lim _{R \rightarrow \infty} \int_{\Gamma} f(z) d z=\frac{\pi}{2} \) \( \Rightarrow \quad \int_{-\infty}^{\infty} f(x) d x+0=\frac{\pi}{2} \Rightarrow \int_{-\infty}^{\infty} f(x) d x=\frac{\pi}{2} \)

Find \( \frac{d y}{d x} \) for \( y=\tan ^{-1}\left\{\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right\} \)

(iv) Find the equation of the tangent to the curne \( y=x^{3}-x \) at \( x=2 \)

\( \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { e ^ { - y } } { y } d y d x \)