I was solving the following problem:
After using the formula $$\text{length} = \int_b^a \sqrt{(x'(t))^2 + (y'(t))^2}\,dt,$$ I got $$\int_0^4 \sqrt{(1)^2+\left(\frac{1}{2}t^{-\frac{1}{2}}\right)^2}\,dt = \int_0^4 \sqrt{1+\frac{1}{4}t^{-1}}\,dt \approx 4.646,$$
which is not one of the answer choices. Can anyone show me my mistake?
I also checked the solution to this problem and it seems that they did not take the derivative as in the formula above:
Is there a reason why the problem was solved in this way? Many thanks in advance for any help.