Use shells, or split the washers at the kink

Find the volume when the area bounded by y=(x−1)/(x+1)y=(x-1)/(x+1), the xx-axis and x=2,x=3x=2,x=3 is rotated completely about the yy-axis. Give your answer to 3 significant figures. marks

[2011/SAJC/Prelim/II/2(a)]

Solution

V

Volume of revolution

Find the volume generated when the area bounded by y=x−1x+1y=\dfrac{x-1}{x+1}, the xx-axis and the lines x=2x=2 and x=3x=3 is rotated completely about the yy-axis. Give your answer to 3 significant figures.

Explore this idea

Hint

  1. Sketch the region: it lies between x=2x=2 and x=3x=3, from the xx-axis up to the curve.

  2. Choose a method. Shells use xx directly: radius xx, height yy. Washers need xx in terms of yy.

  3. For washers, find the heights where the boundary changes: the curve at x=2x=2 and x=3x=3.

Answer · V

V=π[1+4ln⁡(4/3)]≈6.76V=\pi[1+4\ln(4/3)]\approx6.76 cubic units (3 s.f.).

Explore the idea

A thick ring under a gently rising curve

Spin the region round the y-axis. Try washers (horizontal slices) and then shells (vertical tubes). They give the same volume by different routes.

Open the interactive view to explore this geometry in 3D.

Takeaway

Which method?

If the question gives yy as a function of xx and the region is bounded by vertical lines, shells need no inversion and no splitting. Reach for washers when the region is described by horizontal boundaries or when inverting is easy.