Please Show Work

1. Consider the solid region whose base R is bounded by the negative x axis, the positive y axis, and the curve y = 4 - x²for 2 x 0.

a) Let D1 be the solid with base R, and assume that the cross sections of D1 perpendicular to the x axis are squares. Draw a picture of the base, and then draw a representative cross section (perpendicular to the x axis) at some arbitrary x in the interval (2,0). Finally, find the cross-sectional area A1 of the cross section.

b) Now let D2 be the solid with base R, and assume that the cross sections of D2 perpendicular to the x axis are semi-circles. Draw a second picture of the base, and then draw a representative cross section (perpendicular to the x axis) at some arbitrary x in the interval (2,0). Finally, find the cross-sectional area A2 of the cross section.

c) Suppose that the solid D3 had the same base R, but had equilateral triangles for cross sections perpendicular to the x axis. Without necessarily calculating the volumes, order the volumes V1,V2,V3 of D1,D2,D3 from least to greatest.

2. Let f and g be continuous and 0 g(x) f(x) on [a,b]. The washer method yields the volume formula (shown below) when the region between the graphs of f and g is revolved about the x axis. Suppose g(x)

Volume Formula: