The given curve is rotated about the y-axis. Find the area of the resulting surface. x2⁄3 + y2⁄3 = 9, 0 ≤ y ≤ 27

Answer 1

Video recording did not take, so here is a Desmos graph that shows the steps to reach a solution. The idea is to use:

S = 2π ƒxds where ds = √(1+(dy/dx)²)dx

desmos.com/calculator/tto4vrm7ys

The graph shows how to solve the original relation for y in terms of x, how to determine dy/dx and (dy/dx)². Then an expression for ds is determined followed by using the above formula for S (surface area). If the 2nd ds is used, must solve for x in terms of y,

Answer 2

y = (9 - x^2/3)^3/2, 0 ≤ x ≤ 27; dy/dx = 3/2(9 - x^2/3)^1/2(-2/3x^-1/3);

Area of surface: S = 2π∫₀²⁷x√[1 + 9/4(9 - x^2/3)·4/9x^-2/3]dx = 2π∫₀²⁷3x^2/3dx = 18π/5x^5/3₀²⁷= 4374π/5