Arc Length of 1st Full Turn of Archimedean Spiral

In summary, the arc length of 1st full turn of Archimedean spiral is the distance along the spiral curve from the starting point to the end of the first complete revolution. It can be calculated using the formula L = aθ and has various applications in mathematics and science. Compared to other spiral curves, the arc length of 1st full turn of Archimedean spiral is unique in that it is a constant value. Real-world examples of this arc length can be seen in natural and man-made structures such as snail shells, spiral staircases, and sunflower seeds.
  • #1
JeeebeZ
40
1

Homework Statement



27. Use the parametrization x = tcost, y = tsint of the
Archimedean spiral to find the arc length of the first full turn
of this spiral (corresponding to 0 <= t <= 2∏ ).

Homework Equations


The Attempt at a Solution



I use onenote and a tablet, so my exact attempt in in the pdf attached..

View attachment 27.pdf

Im not sure where I went wrong on this one. It looks correct to me.

The answer key shows.

1/2 [ t√(1 + t2) + ln ( t + √{1 + t2})] | 2∏, 0
 
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  • #2
Looks like you made a typo in page 2, second row.
 

Related to Arc Length of 1st Full Turn of Archimedean Spiral

What is the definition of arc length of 1st full turn of Archimedean spiral?

The arc length of 1st full turn of Archimedean spiral is the distance along the spiral curve from the starting point to the end of the first complete revolution.

How is the arc length of 1st full turn of Archimedean spiral calculated?

The arc length of 1st full turn of Archimedean spiral can be calculated using the formula L = aθ, where L is the arc length, a is the distance from the center of the spiral to a point on the curve, and θ is the angle of rotation.

What is the significance of the arc length of 1st full turn of Archimedean spiral in mathematics and science?

The arc length of 1st full turn of Archimedean spiral has many applications in mathematics and science, including in the study of curves and geometric shapes, as well as in the design of spiral-based structures and patterns.

How does the arc length of 1st full turn of Archimedean spiral compare to other spiral curves?

The arc length of 1st full turn of Archimedean spiral is unique in that it is a constant value, whereas the arc length of other spiral curves, such as the logarithmic spiral, increases continuously as the curve rotates.

Are there any real-world examples of the arc length of 1st full turn of Archimedean spiral?

Yes, the arc length of 1st full turn of Archimedean spiral can be seen in various natural and man-made structures, such as the shape of a snail's shell, the design of spiral staircases, and the pattern of sunflower seeds.

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