Third Exam

1. Evaluate the integral.
   1. $${\int_{0}^{2} \int_{0}^{3} (x^{2}+y^{2}) \, dy dx}$$
   2. $${\int_{0}^{1} \int_{0}^{2} (x^{2} y) \, dy dx}$$
2. Sketch the region of integration and evaluate the integral.
   1. $${\int_{1}^{4} \int_{\sqrt{y}}^{y} x^{2} y^{3} \, dx dy}$$
   2. $${\int_{-2}^{0} \int_{-\sqrt{9-x^{2}}}^{0} 2 x y \, dy dx}$$
3. Evaluate the integral by reversing the order of integraion.
   1. $${\int_{0}^{1} \int_{y}^{1} \sin (x^{2}) \, dx dy}$$
   2. $${\int_{0}^{1} \int_{e^{y}}^{e} \frac{x}{\ln x} \, dx dy}$$
4. Evaluate the triple integral of the function over the region $W$ .
   1. $f(x,y,z)=x^{2}+5 y^{2} - z$ , $W$ is the rectangular box $0 \le x \le 2$ , $-1 \le y \le 1$ , $2 \le z \le 3$ .
   2. $f(x,y,z)=\sin x \, \cos(y+z)$ , $W$ is the cube $0 \le x \le \pi$ , $0 \le y \le \pi$ , $0 \le z \le \pi$ .
5. Find the mass of the solid bounded by the $xy$ -plane, $yz$ -plane, $xz$ -plane, and the plane $(x/3)+(y/2)+(z/6)=1$ , if the density of the solid is given by $\delta(x,y,z)=x+y$ .
6. Convert the following integral to polar coordinates and evaluate.
   $$\int_{-1}^{0}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} x \, dy dx$$

7. Evaluate the integral of the function $f(\rho, \theta, \phi) = \sin \phi$ over the region $0 \le \theta \le 2 \pi$ , $0 \le \phi \le \pi/4$ , $1 \le \rho \le 2$ .