Find the volume of the solid obtained by rotating the region

Question

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves

, ,  , about the line .

Answer 1

The region is bounded from above by y = 1,

it is bounded from below and on the right by x = y⁴,

it is bounded on the left by x = 0.

Inside the region, both x and y range between 0 and 1.

At some position y₀ consider a small sub-interval of [0,1] of width dy.

Extending the subinterval horizontally through our region gives a sliver of length y₀⁴.

If that sliver is rotated around the y=1, it is rotated with radius 1-y₀.

Therefore it will generate the volume 2π(1-y₀)y₀⁴dy.

Therefore the volume of the whole region is the sum over all those subintervals dy, which is

∫₀¹ 2π(1-y)y⁴dy =

2π(∫₀¹ y⁴dy - ∫₀¹ y⁵dy) =

2π(1⁵/5 - 1⁶/6) =

π/15 = 0.21