1. Consider the solid created by diagonally slicing a rectangular box with a plane which intersects three of the vertical sides and cuts off one corner of the base. Let the base have length l and width w, and the vertical edges be cut off with heights a, b, and d.
 

(a) Select a convenient coordinate system and find an equation for the plane that slices the box.
 

(b) Find the coordinates for the points where the top plane intersects the bottom plane.
 

(c) Set up an iterated integral and use it to find the volume of the truncated box.
 
 
 
 
 

2. A graph in polar coordinates of the form r = a + bsin for some fixed positive constants a and b is a shape in general called a limaçon. If a<b the graph includes an "inner loop" with size depending on the values of a and b. Find the mass and center of mass of a lamina with constant density occupying the region D which lies between the inner and outer loops of r = a + bsin for any a<b.