Exam 2

1. Calculate the following integrals if they converge; provide a brief explanation if they diverge. (40%)

   1. $$\int_{1}^{\infty} \frac{1}{5 x + 2} \, dx$$

   2. $$\int_{0}^{\infty} x e^{-x^2} \, dx$$

   3. $$\int_{-\infty}^{\infty} \frac{1}{x^{2}+25} \, dx$$

   4. $$\int_{0}^{1} \frac{x^{4}+1}{x} \, dx$$

   5. $$\int_{3}^{\infty} \frac{1}{x (\ln x)^{2}} \, dx$$

2. The gamma function is defined as
   $$\Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} \, dt ,$$   for$$\ x>0 .$$
   Find $\Gamma(2)$ . (10 %)
3. Find the area bounded by the curves $3x+y=6$ , $y=x^2-4$ , and the $y$ axis. (10 %)
4. Find the area bounded by $\sqrt{x}+\sqrt{y}=2$ , the vertical line $x=4$ , and the $x$ axis. (10 %)
5. Find the volume of revolution bounded by $y=\sqrt{x+1}$ , $y=0$ , $x=-1$ , and $x=1$ . (10 %)
6. Write the integral for the arc length for the curve $y=\sqrt{x^{2}-1}$ from $x=2$ to $x=3$ (do not attempt to integrate it!), and use your calculator to estimate the numerical value. (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 %)