Question

How do you do this question?

Answer 1

Answer:

π/10

Step-by-step explanation:

When we rotate the region about the y-axis, we get something that looks like a volcano, or a bundt cake.  Instead of slicing this into flat washers, we'll slice it in concentric rings, or "shells".

Each shell has a radius x, a thickness dx, and a height y.  The volume of an individual shell is:

dV = 2π r h t

dV = 2π x y dx

Since y = x² − x³:

dV = 2π x (x² − x³) dx

dV = 2π (x³ − x⁴) dx

The total volume is the sum of all the shells from x=0 to x=1.

V = ∫ dV

V = ∫₀¹ 2π (x³ − x⁴) dx

V = 2π (¼ x⁴ − ⅕ x⁵) |₀¹

V = 2π (¼ − ⅕)

V = π/10