What is the length of the curve from y = 0 to y = 1?

Question

What is the length of the curve  from y = 0 to y = 1? (4 points)

answer choices

	1
	None of these

Answer 1

Hi Stephanie,

arc length using integration is:

s= ∫_a^b (1 + (dy/dx)² )^1/2 dx

x= (1/3)(y² + 2)^3/2

changing it in terms of y

3x= (y² + 2)^3/2

(3x)^2/3= (y² + 2)

(3x)^2/3 - 2= y²

y = ±((3x)^2/3 - 2)^1/2

we are only considering y>0 so ignore the - part

y = ((3x)^2/3 - 2)^1/2

I used a calculator for this next part

(dy/dx) = (1+[1/(3^(1/3) Sqrt[-2 + 3^(2/3) x^(2/3)] x^(1/3))]^2)^(1/2)

at this point the calculators I used failed so I am going to go with a different approach.

this is getting to complicated for me compute this integral. so Instead I can give an approximation. By making a triangle between the two boundaries. I get a line that can give a really rough approximation. solving for the length of the line gives a result of result of 1.27 which is close to the 4/3

I then split this up into 5 smaller triangles estimating the points and finding the lengths that these triangles approximate and I came up with a length of 1.3344 which is close to 1.3333 so I am assuming that 4/3 is the correct answer but this is just an educated guess.