Calculus II

Lesson 2: Solids of Revolution

Example 1

Find the volume of the solid obtained by rotating the region bounded by the given curves about the

given axis. Sketch the region and a typical disk or washer.

= x, x = 2y; about the y-axis

> restart: with(plottools):with(plots):

Warning, the name changecoords has been redefined

> f:= x -> x/2;

> g1:= x -> sqrt(x);

> g2:= x -> -sqrt(x);

Let's determine where the functions intersect.

> solve( g1(x)=f(x), x);

We have an intersection when x = 4 and y = 2. Let's plot.

> a:=plot(f(x), x = 0..4, color = red, thickness=2):

> b:= plot(g1(x), x = 0..4, color = blue, thickness=2):

> c:= plot(g2(x), x = 0..4, color = blue, thickness=2):

> d:= seq( plot([0 + 4 * i/20, t, t = f( 0 + 4 * i/20)..g1(0 + 4 *i/20)], color = magenta), i = 1..20):

> display({a,b,c,d});

> t1 := tubeplot([0,y,0], y=0..2, radius=y^2, axes=normal, style=wireframe, numpoints=15, tubepoints=50, axes=normal, color=blue):
t2 := tubeplot([0,y,0], y=0..2, radius=2*y, axes=normal, style=wireframe, numpoints=10, tubepoints=50, axes=normal, color=red):
display(t1,t2, orientation=[-90,-10]);

We need to revovle the shaded region about the y-axis. Hence a cross section will be a washer.

> e:= circle([0,0],1,color = brown):

> g:= circle([0,0], 3, color = blue):

> h:= plot([t,0,t = 0..3], color = magenta):

> i:= textplot([2,0.7,`2y`], color = magenta):

> j:= plot([0,t,t = 0..1], color = red, title = `washer`):

> k:= textplot([-0.5,0.5,`y^2`], color = red):

> display({e,g,h,i,j,k}, scaling=constrained);

The area of a washer is given by: (4 ) - ( )

Hence the volume of the solid is precisely:

We have maple do the integration.

> int(Pi * (4 * y^2 - y^4), y = 0..2);

Thus the volume of the solid is cubic units.

>

Example 2

Find the volume of the solid obtained by rotating the region bounded by the given

curve about the x-axis.

y = sec(x), y = 1/2, x = -1, x = 1

Sketch the region. Draw a typical cross setion.

> f:= x -> sec(x);

> a:= plot(f(x), x = -1..1, color = blue):

> b:= plot(1/2, x = -1..1, color = brown, thickness=2):

> c:= seq( plot([-1 + 2*i /30, t, t = 1/2..sec(-1 + 2 * i/30)], color = red), i = 0..30):

> display({a,b,c}, view=[-1..1,0..sec(1)]);

We take the region shaded in red above and revolve it about the x-axis to generate a solid.

A typical cross section will be a washer. We draw a typical cross section.

> t1 := tubeplot([x,0,0], x=-1..1, radius=sec(x), axes=normal, style=wireframe, numpoints=25, tubepoints=80, axes=normal, color=blue):
t2 := tubeplot([x,0,0], x=-1..1, radius=1/2, axes=normal, style=patchnogrid, numpoints=15, tubepoints=30, axes=normal, color=brown):
plane1 := implicitplot3d({x=-1, x=1}, x=-1..1, y=-2..2, z=-2..2, color=grey, style=patchnogrid):
plane1 := implicitplot3d(x=-1, x=-1..1, y=-2..2, z=-2..2, color=grey, style=wireframe, numpoints=100):
plane2 := implicitplot3d(x=1, x=-2..2, y=-2..2, z=-2..2, color=grey, style=wireframe, numpoints=100):
display(t1,t2, plane1, plane2, orientation=[-70,70], scaling=constrained);

> g:= circle([0,0], 3, color = blue):

> h:= circle([0,0], 1, color = blue):

> i:= plot([t,0,t = 0..3], color = magenta, title = `washer`):

> j:= plot([0,t,t = 0..1], color = brown):

> k:= textplot([1.8,.5,`sec(x)`], color = magenta):

> l:= textplot([-0.25,0.5, `1`], color = brown):

> display({g,h,i,j,k,l}, scaling=constrained);

From the plot of a typical cross section, we have the area of a typical cross section is given by:

- ( ) .

Hence the volume of the solid is given by:

2 = 2 .

> int( 2 * Pi * (tan(x))^2, x = 0..1);

Thus the volume of the solid is precisely:

2 ( tan(1) - 1 ) cubic units.

>

Example 3

Use the method of cylindrical shells to compute the volume generated by rotating the

region bounded by the curves about the y-axis.

y = - , y = 0

> g:= x -> x^2 - x^3;

> a:= plot(g(x), x = -0.1..1):

> b:= seq( plot([0 + i/20,t,t = 0..g(0 + i/20)], color = red), i = 1..20):

> c:= plot([t,0.1,t = 0..0.4126055723], color = blue):

> d:= plot([t,0.1,t =0.4126055723..0.8669513176], color = brown):

> e:= textplot([0.2,0.09,`r`], color = blue):

> f:= textplot([0.68,0.12,`R`], color = brown):

> k:= seq(plot([5/20 + (i/10)*(1/20), t, t = 0..g(5/20 + (i/10)*(1/20))], color = magenta), i = 1..10):

> display({a,b, c,d,e,f,k});

We need to rotate the shaded region about the y-axis.

If we want to calculate the volume using washers, we would need to solve the

equation y = - for x. Not a task we want to do. Hence we use

cylindrical shells. Taking the magenta region above and rotating about the y-axis

we get a cylindrical shell.

>

We need to calculate the volume of the above shell. Let be the inner radius and

let be the outer radius. Let h denote the height of the shell. Then the volume is given by:

Volume of shell = h - h

= ( + ) ( - ) h

= 2 h ( - ).

Setting r = - and r = we have

Volume of shell = 2 r h r.

We then integrate over r (or x) to obtain the volume of the solid. .

Volume of solid =

> int(2 * Pi * x * g(x), x = 0..1);

Thus, the volume of the solid is cubic units.

>

Example 4

Use the method of cylindrical shells to find the volume generated by rotating the region

bounded by the given curves about the specified axis.

y = , y = 0, x = 5, about y = 3.

> g:= x -> sqrt(x - 1);

> a:= plot(g(x), x = 1..6, color = brown):

> b:= plot([5,t,t = 0..2], color = magenta):

> c:= plot([t,3,t = 1..6], color = blue):

> d:= seq( plot([t,0 + 2 * i/10, t = (i/5)^2 + 1..5], color = red), i = 1..10):

> e:= seq( plot( [t, 4/5 + i/50, t = (4/5 + i/50)^2 + 1..5], color = blue), i = 1..10):

> display({a,b,c,d,e});

> tubeplot({[x,3,0, x=1..5, radius=3-g(x)], [x,3,0, x=1..5, radius=3]}, scaling=constrained, axes=none, style=wireframe, numpoints=15, tubepoints=30, orientation=[-70,-70], axes=none);

The volume of the solid is given by:

Volume =

> int( 2 * Pi * (3 - y) * (5 - y^2 - 1), y = 0..2);

Thus the volume is 24 cubic units.