MA 2005 A-95 Sample Exam 2 Name here

Show your work in the space provided. Unsupported answers may not receive full credit. Use the backs of the pages if needed. (In the real exam, there will be plenty of space for your work.)

1. Evaluate the following triple integrals.

2. Set up and evaluate a double integral in polar coordinates to find the volume of the solid that lies under the surface z=1+x and above the plane region bounded by the cardioid .

3. Set up and evaluate the integral for the volume of the solid that lies under the surface z=xy and above the region in the xy plane bounded by the curves y=x and . Make sure you include a sketch of the region.

4. For the following integral, first sketch the region of integration, then reverse the order of integration, and finally evaluate the resulting integral.

The region is bounded by the x axis, the line y=x, and the line x=1.

5. Compute the coordinates of the center of mass, , of a thin plate bounded by the x and y coordinate axes and the line x+y=1 if the density per unit area is . Include a sketch of the region.

The region is bounded by x=0, y=0, and the line y=1-x.

6. Use a triple integral to find the volume of the solid bounded by the coordinate planes and the plane 2x+2y+z=2.