Take a cylindrical soup can or vase, and measure its diameter D and height H. Attach to it a string (or a piece of a clothes line wire), forming a spiral from the bottom to the top, going around exactly once. Compute its length using an integral formula, and compare this value to the measured length of the piece of string used.

View of a string around a can

t is an angle measured in degrees around the axis of symmetry of a can.

x = D/2*cos(t)^{ } _{ } |
one horizontal coordinate;^{ } _{ } |
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y = D/2*sin(t)^{ } _{ } |
another horizontal coordinate;^{ } _{ } |
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z = H/360*t^{ } _{ } |
vertical coordinate.^{ } _{ } |

Look at the

One full turn of 360 degrees lifts the string to the height H.

Length = Integral of √((dx/dt)

Set MODE to DEGREE and PAR.

Define,

\X1T=D/2cos(T)^{ } _{ } |
this is x;^{ } _{ } |
|||

\Y1T=D/2sin(T)^{ } _{ } |
this is y;^{ } _{ } |
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\X2T=H/360T^{ } _{ } |
this is z.^{ } _{ } |

Enter values of D and H.

Compute,

Y1(360)^{ } _{ } |

ENTER^{ } _{ } |

Another question: How long is a string that goes around the can TWICE?

You can compute this using an integral, or using the method shown in Part 2.

Part 2.

Finding the length of the string using the Pythagorean Theorem

Cut a rectangular piece of paper to just cover the lateral surface of your cylindrical soup (or other) can or vase. Tape the paper on the can. Next, measure the can's diameter D and height H. Attach to the can a string (or a piece of clothes line wire), and form a spiral from the bottom to the top, going around exactly once. With a pencil, draw the spiral (it is called a helix) on the paper that is wrapped around the can. Now untape the paper and lay it flat. What shape does the line make now? See the pictures below!

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