Question

Consider the path r(t)=(10t,5t2,5lnt) defined for t>0. Find the length of the curve between the points (10,5,0) and (30,45,5ln(3)).

Answer 1

The length of the curve $C$ is

$\displaystyle\int_C\mathrm dS=\int_1^3\left|\frac{\mathrm d\vec r(t)}{\mathrm dt}\right|\,\mathrm dt$

$=\displaystyle\int_1^3\sqrt{\left(\frac{\mathrm d(10t)}{\mathrm dt}\right)^2+\left(\frac{\mathrm d(5t^2)}{\mathrm dt}\right)^2+\left(\frac{\mathrm d(5\ln t)}{\mathrm dt}\right)^2}\,\mathrm dt$

$=\displaystyle\int_1^3\sqrt{100+100t^2+\frac{25}{t^2}}\,\mathrm dt$

$=\displaystyle\int_1^3\sqrt{\frac{25}{t^2}}\sqrt{4t^2+4t^4+1}\,\mathrm dt$

We have $\sqrt{t^2}=|t|$, but integrating over [1, 3] makes $t>0$, so $|t|=t$. Under the square root, we can factorize

$4t^4+4t^2+1=(2t^2+1)^2$

and for the same reason as before, $\sqrt{(2t^2+1)^2}=2t^2+1$

Then the integral is

$=\displaystyle5\int_1^3\frac{2t^2+1}t\,\mathrm dt$

$=\displaystyle5\int_1^3\left(2t+\frac1t\right)\,\mathrm dt$

$=5(t^2+\ln t)\bigg|_1^3=\boxed{40+5\ln3}$

Answer 2

The length of the curve is of: $\mathbf{\frac{70}{3}}$ units.

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The length of a parametric curve $r(t) = (x(t), y(t), z(t))$ in an interval [a,b] is given by:

$L = \int_{a}^{b} \sqrt{x^{\prime}(t)^2 + y^{\prime}(t)^2 + z^{\prime}(t)^2} dt$

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The curve is:

$r(t) = (10t, 5t^2 + 5\ln{t}}$

Thus:

$x(t) = 10t, x^{\prime}(t) = 10$

$y(t) = 5t^2, y^{\prime}(t) = 10t$

$z(t) = 5\ln{t}, z^{\prime}(t) = \frac{5}{t}$

Between points $(10,5,0) \rightarrow a = 1$ and $(30, 45, 5\ln{3}) \rightarrow b = 3$

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The length is:

$L = \int_{a}^{b} \sqrt{x^{\prime}(t)^2 + y^{\prime}(t)^2 + z^{\prime}(t)^2} dt$

$L = \int_{1}^{3} \sqrt{100 + 100t^2 + \frac{25}{t^2}} dt$

$L = \int_{1}^3 \sqrt{\frac{100t^4 + 100t^2 + 25}{t^2}} dt$

$L = \int_{1}^3 \sqrt{\frac{(10t^2 + 5)^2}{t^2}} dt$

$L = \int_1^3 \frac{10t^2 + 5}{t^2} dt$

$L = \int_1^3 10 + 5t^{-2} dt$

$L = 10t - \frac{5}{t}|_{1}^{3}$

$L = 30 - \frac{5}{3} - 10 + 5$

$L = 25 - \frac{5}{3} = \frac{75}{3} - \frac{5}{3} = \frac{70}{3}$

The length of the curve is of $\mathbf{\frac{70}{3}}$ units.

A similar problem is given at https://brainly.com/question/11089689