Second Exam

1. Evaluate the following integrals if they converge; provide a brief explanation if they diverge. (40%)
   1. $${ \int_{1}^{\infty} \frac{1}{6 x + 1} \, dx }$$
   2. $${ \int_{0}^{\infty} x e^{-x^2} \, dx }$$
   3. $${ \int_{-\infty}^{\infty} \frac{1}{x^{2}+9} \, dx }$$
   4. $${ \int_{3}^{\infty} \frac{1}{x (\ln x)^{2}} \, dx }$$

2. Find the area bounded by the curves $y=6-3x$ , $y=x^2-4$ , and the $y$ axis. (10 %)
   

3. Find the area bounded by $\sqrt{x}+\sqrt{y}=\sqrt{2}$ , the $y$ -axis and the $x$ -axis. (10 %)
   

4. Find the volume of revolution bounded by $y=\sqrt{x+2}$ , $y=0$ , $x=-2$ , and $x=2$ . (10 %)
   

5. Find the arc length for the curve $f(x)=\sqrt{x^{3}}$ from $x=0$ to $x=2$ . (10 %)

6. Find the arc length of $${y=\frac{x^{3}}{6} + \frac{1}{2x}}$$ for $\frac{1}{2} \le x \le 1$ . (10 %)

7. Find the volume of a solid formed by rotating the curve $y=1/x$ for $1 \le x < \infty$ around the $x$ axis. (10 %)

8. The force, $F$ , required to compress a spring by a distance $x$ meters is given by $F=3x$ newtons. Find the work done in compressing the spring from $x=0$ to $x=2$ . (10 %)