Let R be the region bounded by y=2sin(π2x)\displaystyle {y}={2}{\sin{{\left(\frac{\pi}{{2}}{x}\right)}}}, y=2(x2)2\displaystyle {y}={2}{\left({x}-{2}\right)}^{{2}}, and y=x+1\displaystyle {y}={x}+{1}, and containing the point (2,2).

a) Find the area of R

 

b) Find the volume of R rotated about the x-axis

 

c) Find the volume of R rotated about the y-axis

 

d) Suppose R is the base of a shape in which cross-sections perpendicular to the x-axis are squares. Find the volume of this shape.