Question

Find the arc length function for the curve y= integration 1 to x √(t^3-1)dt,1<x<4?

Answer 1

The arc length is `62/5` units.

Explanation:

I'm assuming you want us to find the arc length of `int_1^x sqrt(t^3 - 1)dt` on `[1, 4]`.

Recall that `d/dx(int_1^x F(t) dt) = F(x)` and that arc length is given by `A = int_a^b sqrt(1 + F(x)^2) dx`

Then we get

`A = int_1^4 sqrt(1 + (sqrt(x^3 - 1)^2)) dx`

`A = int_1^4 sqrt(x^3) dx`

`A = int_1^4 x^(3/2) dx`

`A = [2/5x^(5/2)]_1^4`

`A = 2/5(4)^(5/2) - 2/5(1)^(5/2)`

`A = 64/5 - 2/5 = 62/5`

Hopefully this helps!