shell method integration

Question

Use the shell method to find the exact value of the volume formed by revolving the region bounded by the graphs of , ,  and  about the y-axis.

Answer 1

As we are rotating around the y-axis, we integrate with respect to x:

V = ∫_a^b 2π·r(x)h(x) dx

--> Bounds: [a,b] = [0,³√π]

--> Radius: r(x) = x

--> Height: h(x) = f(x) = x·sin(x³)

V = ∫₀^{π^(1/3)} 2π⋅x⋅xsin(x³) dx

V = 2π∫₀^{π^(1/3)} x² sin(x³) dx

u = x³, du = 3x²dx, (1/3)du = x²dx

V = 2π⋅(1/3)∫₀^{π^(1/3)} sin(u)du

V = (2π/3)[-cos(u)] [0,³√π]

V = (2π/3)[-cos(x³)] [0,³√π]

V = (2π/3)[-cos((³√π)³) - (-cos(0³))]

V = (2π/3)[-cos(π) + cos(0)]

V = (2π/3)[-(-1) + 1]

V = (2π/3)(1+1)

V = (2π/3)(2)

V = 4π/3 cubic units

Hope this helped!