Length of Arc | MATHalino reviewer about Length of Arc

Problem
Given the position function x(t) = t⁴ - 8t², find the distance that the particle travels at t = 0 to t = 4.

A. 160	C. 140
B. 150	D. 130

Read more about The Distance the Particle Travels with Given Position Function x(t) = t^4 - 8t^2
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Problem
What is the perimeter of the curve r = 4(1 + sin θ)?

The answer is 32 units. For detailed solution, follow the link by clicking on the figure.

Read more about Perimeter of the curve $r = 4(1 + \sin \theta)$ by integration
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The length of arc in polar plane is given by the formula:

$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$

The formula above is derived in two ways.

Read more about Length of Arc in Polar Plane | Applications of Integration
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The length of arc in rectangular coordinates is given by the following formulas:

$\displaystyle s = \int_{x_1}^{x_2} \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$ and $\displaystyle s = \int_{y_1}^{y_2} \sqrt{1 + \left(\dfrac{dx}{dy} \right)^2} \, dy$

See the derivations here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-xy-plane…

Read more about Length of Arc in XY-Plane | Applications of Integration
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Problem
Chords AB and AC are drawn on a circle of radius 10 inches. Find the angle between the chords if the arc BAC is 28 inches long.

Read more about Angle between two chords
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Problem
A circle of radius r rolls along a horizontal line without skidding.

Find the equation traced by a point on the circumference of the circle.
Determine the length of one arc of the curve.
Calculate the area bounded by one arc of the curve and the horizontal line.

Read more about Cycloid: equation, length of arc, area
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Problem 717
Locate the centroid of the bent wire shown in Fig. P-717. The wire is homogeneous and of uniform cross-section.

Read more about 717 Symmetrical Arcs and a Line | Centroid of Composite Line
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The total surface area of the sphere is four times the area of great circle. To know more about great circle, see properties of a sphere. Given the radius r of the sphere, the total surface area is

Read more about Derivation of Formula for Total Surface Area of the Sphere by Integration
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