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Chapter 8: Problem 49

Give the integral formulas for area and arc length in polar coordinates.

Short Answer

Expert verified
The integral formula for area in Polar Coordinates is \[\int_{a}^{b} \frac{1}{2} {r(\theta)}^{2} \, d\theta\] and for arc length is \[\int_{a}^{b} \sqrt{(r(\theta))^2 + \left(\frac{dr}{d\theta}\right)^2} \,d\theta\].

Step by step solution

01

Formulate the Integral Formula for Area in Polar Coordinates

To find the area between two radii, the integral formula in polar coordinates is: \[\int_{a}^{b} \frac{1}{2} {r(\theta)}^{2} \,d\theta\] where \(a\) and \(b\) are the limits of integration, \(r(\theta)\) is the polar function, and \(\theta\) is the angle.
02

Formulate the Integral Formula for Arc Length in Polar Coordinates

To find the arc length, the integral formula in polar coordinates is: \[\int_{a}^{b} \sqrt{(r(\theta))^2 + \left(\frac{dr}{d\theta}\right)^2} \,d\theta\] where \(a\) and \(b\) are the limits of integration, \(r(\theta)\) is the polar function and \(\frac{dr}{d\theta}\) is its derivative with respect to \(\theta\).

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Most popular questions from this chapter

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sec \theta, \quad y=3 \tan \theta $$

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta} .\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\).

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 \(x\) -axis and (b) the \(y\) -axis.

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \sin \theta}\)

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? (a) \(x=t+1, y=t^{3}\) (b) \(x=-t+1, y=(-t)^{3}\)
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