1. Sketch the region between y = e^x , x = 0, x = 2 and y = 0. As well, find the volume of S, where S is the solid obtained by rotating this region about the line y = 0.

2. A curve on the xy-plane passes through the point (0, a) for some number a, and has the property that the slope of the curve at every point P is three times the y-coordinate of P. Find the equation of the curve.

3. Let H(t) be the amount of raw honey at time t. During processing, the amount of raw honey present decreases at a rate proportional to H. In the first ten hours, five hundred pounds of raw honey has been reduced to four hundred pounds. Find how many pounds of raw honey remain after 10 more hours of processing.

4. Briefly explain what is mathematical induction. Then present a differentiation or integration equation that requires mathematical induction to prove, and show the details of this proof. By differentiation equation, this means an equation that has a derivative in it. By integration equation, this means an equation that has an integral in it.