Find the volume of a solid when rotating by the X axis and Y axis respectively.

Question

Suppose that R is the finite region bounded by f(x)=3√x and f(x)=x

Find the exact value of the volume of the object we obtain when rotating R about the x-axis

Vx = ?

Now do the same for the y-axis

Vy = ?

Answer 1

Intersection points x-coordinates: 3√x = x; 9x = x²; x(9 - x) = 0; x = 0, x = 9.

When rotaition about x-axis by Disk Method: V_x = π∫₀⁹[(3√x)² - x²]dx = π(9x²/2 - x³/3)₀⁹ = 243π/2 = 381.7

When rotation abot y-axis then by Shell Method: V_y = 2π∫₀⁹x(3√x - x)dx = 2π∫₀⁹(3x^3/2 - x²)dx =

2π(6/5x^5/2- x³/3)₀⁹ = 486π/5 = 305.4

Answer 2

Hi,

Vx=121.5*Pi

Vy=97.2*Pi

Send me a message so I can send you pictures of the work I did.

Hope this helps.