Find the Volume of the Solid

Question

**Please show work

Find the volume of the solid obtained by rotating the region bounded by y=4sin⁡(7x2),y=0,0≤x≤sqrt(pi/7), about the y axis.

Volume : ?

Answer 1

Shell Method (Vertical Axis)

V = 2π∫[a,b] r(x)h(x)dx

Height: h(x) = 4sin(7x²)

Radius: r(x) = x

Bounds: [a,b] = [0,√(π/7)]

Set up integral and evaluate

V = 2π∫[0,√(π/7)] x(4sin(7x²))dx

V = 8π∫[0,√(π/7)] xsin(7x²))dx

Let u=7x² and du=14xdx so that du/14 = xdx

Bounds become u=7(0)²=0 to u=7(√(π/7))²=π:

V = 8π(1/14)∫[0,π] sin(u)du

V = (4π/7)∫[0,π] sin(u)du

V = (4π/7)[-cos(u)] [0,π]

V = (4π/7)[-cos(π)] - (4π/7)[-cos(0)]

V = 4π/7 + 4π/7

V = 8π/7

Therefore, the volume of the solid obtained is 8π/7 cubic units.

Hope this helped!