Example 209. Find the area \( A \) of the region formed by the curve \( y=\frac{1}{x^{2}} \), the line \( x=1 \) and the \( x \)-axis. Proof. The area \( A \) is

Evaluate the double integral \[ I=\iint_{A} 18 y e^{x^{2}} d x d y \] when \( A \) is the region in the first quadrant bounded by the graphs of \[ x=y^{2}, \quad x=2, \quad y=0 \] 1. \( I=18\left(e^{4}+1\right) \) 2. \( I=\frac{9}{2}\left(e^{4}-1\right) \) 3. \( I=9\left(e^{4}-1\right) \) 4. \( I=\frac{9}{2} e^{4} \) 5. \( I=\frac{9}{2}\left(e^{4}+1\right) \)

The following function is given. \[ f(x)=3 x^{3}-7 x^{2}-75 x+175 \]

\( f^{\prime}(x)=\frac{x^{2}-4}{2 x^{2}} \) colculer \( f^{\prime \prime}|x| \)

a) \( \lim _{x \rightarrow-2} \frac{x^{2}+3 x+6}{(x+2)^{2}} \)

\[ f^{\prime \prime}(x)=\frac{4}{x^{3}} \] etudiar la concuvite