1. Area of R = [g(x) – f(x)] dx
Part (a)
Area of R = [g(x) – f(x)] dx
???
Area of R = [4-x – (1/4 + sin (px))] dx
???
.178
.178
Area of R = .064 or.065 units2
Using a graphing calculator, we can find that the curves first intersect at x =.178

2. Area of S = [f(x) – g(x)] dx
Part (b)
Area of S = [f(x) – g(x)] dx
.178 1
Area of S = [(1/4 + sin (px)) - 4-x ] dx
.178 1
Area of S = .410 units2
Using a graphing calculator, we can find that the curves also intersect at x = 1.

3. To find this volume, we’ll need to use the WASHER method.
Part (c)
Volume = or units3
To find this volume, we’ll need to use the WASHER method.
Outer radius = f(x)+1
Inner radius = g(x)+1
y = -1
Volume = p ([f(x)] [g(x)]2) dx
.178 1
[¼+sin(px)+1 ]2
[4-x+1]2