Question

Set up integral for curve length x=sqrt(y)-y [1,4]
I don't have to solve by hand, just set up and solve by calculator. I got 3.1008 but book says it should be 3.6095. Any ideas?

Answer 1

Is the domain \([1,4]\) for \(x\) or \(y\) ?

Answer 2

\[\begin{align*}L&=\int_CdS\\\\
&=\int_1^4\sqrt{1+\left[\frac{dx}{dy}\right]^2}~dy
\end{align*}\]
where \(\dfrac{dx}{dy}=\dfrac{1}{2\sqrt y}-1\), so the integral would be
\[\begin{align*}L&=\int_1^4\sqrt{1+\left[\frac{1}{2\sqrt y}-1\right]^2}~dy\\\\
&=\int_1^4\sqrt{2+\frac{1}{4y}-\frac{1}{\sqrt y}}~dy
\end{align*}\]
Computed with Mathematica, I'm getting the right answer.

Answer 3

sorry, yes the domain is for x