/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Arc Length in Polar Form Use the... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 85

Arc Length in Polar Form Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Short Answer

Expert verified
The formula for the arc length of a polar curve is \(L = \int_a^b \sqrt{(dr/d胃)^2 + r^2} d胃 \)

Step by step solution

01

Write down the formula for arc length in parametric form

Arc length \(L\) in parametric form is given by the formula: \[ L = \int_a^b \sqrt{ (dx/dt)^2 + (dy/dt)^2} dt\] where \(x(t)\) and \(y(t)\) are parametric equations.
02

Convert to polar form

In polar form we have \( x = r cos 胃 \) and \( y = r sin 胃 \) where \( r = r(胃) \), we can get \[ dx/dt = dr/d胃 cos 胃 鈭� r sin 胃 d胃/d胃 \] \[ dy/dt = dr/d胃 sin 胃 + r cos 胃 d胃/d胃 \]
03

Substitute into the arc length formula

Substitute \(dx/dt\) and \(dy/dt\) into the arc length formula:\[ L = \int_a^b \sqrt{[(dr/d胃 cos 胃 - r sin 胃) ^2 + (dr/d胃 sin 胃 + r cos 胃)^2]} d胃 \]. Simplify under the square root:
04

Simplify the equation

Now, simplify the equation to derive the polar form. This is done by expansion and combination of like terms:\[L = \int_a^b \sqrt{(dr/d胃)^2 + r^2} d胃 \]
05

Final formula for arc length in polar form

Finally, the simplified expression represents the formula for the arc length of a polar curve:\[L = \int_a^b \sqrt{(dr/d胃)^2 + r^2} d胃 \]

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Most popular questions from this chapter

Finding the Area of a Polar Region In Exercises \(17-24\) , use a graphing utility to graph the polar equation. Find the area of the given region analytically. Between the loops of \(r=2(1+2 \sin \theta)\)

Area of a Surface of Revolution Give the integral formulas for the area of the surface of revolution formed when the graph of \(r=f(\theta)\) is revolved about (a) the polar axis. (b) the line \(\theta=\pi / 2\) .

Finding the Area of a Polar Region Between Two Curves In Exercises \(43-46,\) find the area of the region. Inside \(r=2 a \cos \theta\) and outside \(r=a\)

Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{10}{5+4 \sin \theta} $$

Finding the Arc Length of a Polar Curve In Exercises \(57-62,\) use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\ {r=2 \sin (2 \cos \theta)} & {0 \leq \theta \leq \pi}\end{array} $$
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