APFIVE Parametric Arc Length Integral Setup
Given the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=\sqrt{t}$$ for $$t\in [1,e^2]$$, which of the following integrals correctly represents the arc length of the curve?
| t | x(t)=ln(t) | y(t)=√t |
|---|---|---|
| 1 | 0 | 1 |
| e | 1 | √e |
| e^2 | 2 | e |
A
$$\int_{1}^{e^2} \left(\frac{1}{t}+\frac{1}{2\sqrt{t}}\right)dt$$
B
$$\int_{1}^{e^2} \sqrt{\frac{1}{t}+\frac{1}{4*t}}\,dt$$
C
$$\frac{1}{2}\int_{1}^{e^2} \sqrt{\frac{1}{t^2}+\frac{1}{4*t}}\,dt$$
D
$$\int_{1}^{e^2} \sqrt{\frac{1}{t^2}+\frac{1}{4*t}}\,dt$$
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