Answer:
π/49·(44 -π)
Step-by-step explanation:
Using the "shell" method, a differential of volume is the product of the area of a cylindrical shell and its circumference around the axis of rotation. Here, that area is ...
dA = y·dx = cos(7x)dx
The radius will be the difference between x and the axis of rotation, x=3, so is ...
r = 3 -x
Then the differential of volume is ...
dV = 2πr·dA = 2π(3-x)cos(7x)dx
The volume will be the integral of this over the limits x ∈ [0, π/14].
∫dV = 6π·∫cos(7x)dx -2π·∫x·cos(7x)dx . . . from 0 to π/14
= (6/7)π·sin(7x) -(2/49)π·(cos(7x) +7x·sin(7x)) . . . from 0 to π/14
= (6/7)π(1 -0) -(2/49)π((0 -1) +(π/2-0))
= π(42/49 +2/49 -π/49)
= (π/49)(44 -π) . . . . cubic units . . . . . approx 2.61960 cubic units
