Line integral computing methods

Question

Evaluate $$\int_L \frac{ds}{\sqrt{x^2+y^2+4}}$$ where $L$ is a straight line segment between points $A(0;0)$ and $B(1;2)$ Can anyone give me a hint how to solve this integral?

Answer 1

With the paramrtrization $(x(t),y(t))=(t,2t)$ with $t \in [0,1]$ we get $||(x'(t),y'(t)||=\sqrt{5}$ and therefore

$\int_L \frac{ds}{\sqrt{x^2+y^2+4}}=\int_0^1 \frac{\sqrt{5}}{\sqrt{4+5t^2}}dt$.

Can you proceed ?

Answer 2

Along the segment $AB$ the relationship reads $y = 2 x$ then

$$ ds=\sqrt{dx^2+dy^2}=\sqrt{1+4}dx $$

and then the integral is

$$ \int_{0}^{1}\frac{\sqrt 5}{\sqrt{5x^2+4}}dx $$