How do you find the volume of the solid obtained by revolving the curve given by #x=3cos^3(t)#, #y=5sin^3(t)# about the #x#-axis?

Question

Scarlett Allison · Accepted Answer

#y^2=25sin^6t=25(1-cos^2t)^3=25[1-(x/3)^{2/3}]^3# By Disk Method, #V=pi int_{-3}^3y^2 dx=25piint_{-3}^3[1-(x/3)^{2/3}]^3dx#

Paisley Johnson · Answer

undefined To find the volume of the solid obtained by revolving the curve x = 3cos^3(t), y = 5sin^3(t) about the x-axis, you can use the method of cylindrical shells or disk/washer method. With cylindrical shells, you integrate the circumference of each shell multiplied by its height. With the disk/washer method, you integrate the area of each disk or washer created by revolving the curve.

Let's use the disk/washer method:

1. Determine the limits of integration for t.
2. Express the volume element dV in terms of t.
3. Integrate dV over the appropriate interval.

For this curve, t ranges from 0 to π/2.

The volume element dV is given by dV = πy^2 dx, where y = 5sin^3(t) and dx = -9cos^2(t)sin(t) dt.

Integrate dV from t = 0 to t = π/2 to find the total volume of the solid.

V = ∫[0,π/2] π(5sin^3(t))^2 (-9cos^2(t)sin(t)) dt