Let f and g be the functions given by f(x) = 1+sin(2x) and g(x) = eˆx/2. Let R be the shaded region in the first quadrant enclosed by the graphs of f and g as shown in the figure above.

a. Find the area of R.

b. The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles with diameters extending from y = f(x) to y = g(x). Find the volume of this solid.

I am not asking anyone to answer these two questions; I have solved them and I got the correct answers, but I don't understand why my answers are correct. The answer to part a is BIGGER than the answer to part b. How can this be true? The area of the base can't be bigger than the volume of the solid??

Please help. Thanks