Find n and this volume, both in exact form.

Question

Consider the trapezoidal region bounded by the graphs of x = 0, x = 1, y = 0, and y = x + n for a certain positive real number n. The volume obtained by revolving this region about the x-axis is equal to the volume obtained by revolving this region about the y-axis. Find n and this volume, both in exact form.

Answer 1

The volume when rotating about the y axis is

∫₀¹ 2π x f(x) dx =

∫₀¹ 2π x (n + x) dx =

n π x² + 2/3 π x³ |₀¹ =

n π + 2/3 π

The volume when rotating about the x axis is

∫₀¹ π f²(x) dx =

∫₀¹ π (n+x)² dx =

π (n+x)³/3 |₀¹ =

π (n+1)³/3 - π n³/3 =

πn² + πn + π/3

Then, if the 2 volumes are alike,

n π + 2/3 π = πn² + πn + π/3, or

1/3 π = πn² or

1/3 = n²

As n is positive, n = √(1/3) = √3/3

Then, the volume is n π + 2/3 π = √3/3 π + 2/3 π = 3.9082