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?
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Parameterize x and y in terms of t. Parameterize t in terms of s. Bring it all together, stir, and serve. – Mateen Ulhaq Jun 27 '18 at 11:11
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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 ?
Fred
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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
$$ \int_{0}^{1}\frac{\sqrt 5}{\sqrt{5x^2+4}}dx $$
Cesareo
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This is exactly how I solved and my problem actually started right here. The only thing I can think about is $u$ substitution, but it will not work because there is no extra $x$ in integrand since $du=10xdx$ – Martins2018 Jun 27 '18 at 11:24
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@Martins2018 first take $\sqrt{5}$ common from denominator. Then you'll be able to use direct formula to evaluate the integral – So Lo Jun 27 '18 at 11:31