Math Problems Vault

Problem 3366

Find the arc length of the curve $y=\sqrt{x}$ for $0<x<1$ .

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Problem 3367

Find the arc length of the curve $y=\ln x$ for $\sqrt{3}<x<\sqrt{8}$ .

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Problem 3368

Find the arc length of the curve $y=a\ln \cos \frac{x}{a}$ for $0<x<b$ .

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Problem 3369

Find the arc length of the curve $y=a\ch \frac{x}{a}$ for $0<x<x_{0}$ .

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Problem 3370

Find the arc length of the curve $y=\frac{x-3a}{3}\sqrt{\frac{x}{a}}$ for $0<x<x_{0}$ .

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Problem 3371

Find the arc length of the curve $y=2a\ln \frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}} - 4\sqrt{ax}$ for $0<x<x_{0}$ .

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Problem 3372

Find the arc length of the curve $x=a\ln \frac{a+\sqrt{a^{2}-y^{2}}}{y} - \sqrt{a^{2}-y^{2}}$ for $b<y<a$ .

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Problem 3373

Find the arc length of the curve $y^{3}=px^{2}$ for $0<y<y_{0}$ .

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Problem 3374

Find the arc length of the curve $(2a-x)y^{2}=x^{3}$ for $0<x<x_{0}$ , where $x_{0}<2a$ .

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Problem 3375

Find the arc length of the curve $x^{2/3}+y^{2/3}=a^{2/3}$ . The full perimeter.

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Problem 3376

Find the arc length of the curve $x=a\cos^{5}t$ , $y=a\sin^{5}t$ . The full perimeter.

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Problem 3377

Find the arc length of the curve $x=a(t-\sin t)$ , $y=a(1-\cos t)$ . One arch.

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Problem 3378

Find the arc length of the curve $r=a(1+\cos \varphi )$ . The full perimeter.

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Problem 3379

Find the arc length of the first turn of the Archimedean spiral $r=a\varphi$ .

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Problem 3380

Find the arc length of the logarithmic spiral $r=ae^{m\varphi }$ , $0<r<a$ .

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Problem 3381

Prove that the length of one arch of the epicycloid

x = a\left((n+1)\cos t - \cos (n+1)t\right), \qquad y = a\left((n+1)\sin t - \sin (n+1)t\right)

equals $8a\frac{n+1}{n}$ .

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Problem 3382

Prove that the length of one arch of the hypocycloid

x = a\left((n-1)\cos t + \cos (n-1)t\right), \qquad y = a\left((n-1)\sin t - \sin (n-1)t\right)

equals $8a\frac{n-1}{n}$ .

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Problem 3383

Determine the arc length of the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ between its points of contact with the coordinate axes.

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Problem 3384

Find the length of the entire curve $\left(\frac{x}{a}\right)^{2/3}+\left(\frac{y}{b}\right)^{2/3}=1$ .

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Problem 3385

Find the arc length of the curve

x=8at^{3}, \qquad y=3a(2t^{2}-t^{4}),

lying above the $Ox$ axis.

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Problem 3386

The parabola $4ay=x^{2}$ rolls along the $Ox$ axis. Prove that its focus traces out the catenary $y=a\ch \frac{x}{a}$ .

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Problem 3387

Prove that when the cardioid $r=a(1-\sin \varphi )$ rolls along the cycloid

x=a(t-\sin t), \qquad y=a(1-\cos t)

its cusp traces out a straight line.

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Problem 3388

Find the arc lengths of the following space curves: $x=a\cos t$ , $y=a\sin t$ , $z=bt$ for $0<t<t_{0}$ .

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Problem 3389

Find the arc length of the space curve $x=e^{t}\cos t$ , $y=e^{t}\sin t$ , $z=e^{t}$ for $-\infty <t<0$ .

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Problem 3390

Find the arc length of the space curve $x=at$ , $y=\sqrt{3ab}\, t^{2}$ , $z=2bt^{3}$ for $0<t<t_{0}$ .

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Problem 3391

Find the arc length of the space curve $x^{2}=3y$ , $2xy=9z$ for $0<x<x_{0}$ .

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Problem 3392

Find the arc length of the space curve $y=\arcsin \frac{x}{a}$ , $z=\frac{a}{4}\ln \frac{a+x}{a-x}$ for $0<x<x_{0}$ .

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Problem 3393

Find the arc length of the space curve $y^{2}=2ax-x^{2}$ , $z=-a\ln \left(1-\frac{x}{2a}\right)$ ; $0<x<x_{0}$ .

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Problem 3394

Find the arc length of the space curve $4ax=(y+z)^{2}$ , $4x^{2}+3y^{2}=3z^{2}$ ; $0<x<x_{0}$ .

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Problem 3395

Find the arc length of the space curve $(y-z)^{2}=3a(y+z)$ ; $9x^{2}+8y^{2}=8z^{2}$ ; $0<x<x_{0}$ .

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Problem 3396

Find the arc length of the space curve $x^{2}+y^{2}=az$ , $y=x\tg \frac{z}{a}$ .

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Problem 3397

Find the arc length of the space curve $x^{2}+y^{2}+z^{2}=a^{2}$ , $\sqrt{x^{2}+y^{2}}\ch \arctg \frac{y}{x}=a$ .

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