Answer:
V = 8π²/3 − 2π√3
Step-by-step explanation:
Start by drawing the region. Evaluate the function at the endpoints.
f(x) = y = 1 + sec(x)
f(-π/3) = 1 + sec(-π/3) = 3
f(π/3) = 1 + sec(π/3) = 3
Therefore, the curve is the one shown in the left side options. We want the region between the curve and y=3, so the correct sketch is the lower left.
When we revolve this region about y=1, we get a hollow cylinder on its side (again, the answer in the lower left).
Slicing vertically, we get washers with thickness dx, outside radius of 3 − 1 = 2, and inside radius of 1 + sec(x) − 1 = sec(x). The volume of the washer is:
dV = π (R² − r²) t
dV = π (2² − sec²(x)) dx
dV = π (4 − sec²(x)) dx
The total volume is the sum of all the washers from x=-π/3 to x=π/3.
V = ∫ dV
V = ∫ π (4 − sec²(x)) dx
V = π (4x − tan(x))
Evaluate from x=-π/3 to x=π/3.
V = π (4(π/3) − tan(π/3)) − π (4(-π/3) − tan(-π/3))
V = π (4π/3 − √3) − π (-4π/3 + √3)
V = 4π²/3 − π√3 + 4π²/3 − π√3
V = 8π²/3 − 2π√3
