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Chapter 12: Problem 63

Find the length of the following polar curves. The complete circle \(r=a \sin \theta,\) where \(a>0\)

Short Answer

Expert verified
Answer: The length of the polar curve is \(L = 2\pi a\).

Step by step solution

01

Find the derivative of r with respect to 胃

The polar equation is given by \(r = a\sin\theta\). To find its derivative with respect to \(\theta\), we'll use the chain rule: \(\frac{dr}{d\theta} = a\cos\theta\)
02

Determine the bounds of integration

Since the polar equation represents a complete circle, it will cover the range from \(\theta = 0\) to \(\theta = 2\pi\).
03

Apply the polar arc length formula

The polar arc length formula is given by \(L = \int_\alpha^\beta\sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}d\theta\) Plugging in the values for \(r\) and \(\frac{dr}{d\theta}\), we get: \(L = \int_0^{2\pi}\sqrt{(a\sin\theta)^2 + (a\cos\theta)^2}d\theta\)
04

Simplify and integrate

The integrand can be simplified by factoring out \(a^2\): \(L = \int_0^{2\pi}\sqrt{a^2(\sin^2\theta + \cos^2\theta)}d\theta\) Recall that \(\sin^2\theta + \cos^2\theta = 1\). Then, \(L = \int_0^{2\pi}\sqrt{a^2}d\theta\) \(L = a\int_0^{2\pi}d\theta\) Now, integrate: \(L = a[\theta]_0^{2\pi} = a(2\pi - 0)\)
05

Write the final result

The length of the polar curve is \( L = 2\pi a \) where \(a>0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arc Length
When working with polar curves, finding the arc length involves a specific formula that helps to measure the distance along the curve. The arc length of a curve in polar coordinates, such as a circle or spiral, can be calculated using:
  • the formula: \( L = \int_\alpha^\beta\sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}d\theta \)
  • where \( r \) is the radial coordinate representing the curve's distance from the origin.
  • \( \alpha \) and \( \beta \) are the limits of integration, representing the starting and ending angles.
To apply this formula, you must determine \( r \) and its derivative \( \frac{dr}{d\theta} \). The task involves substituting these into the formula and integrating over the angle. This process efficiently calculates the arc length by assessing the sum of all infinitesimal linear segments that form the entire curve.
Chain Rule
The chain rule is a fundamental technique in calculus, used here to find the derivative of \( r = a\sin\theta \) with respect to \( \theta \). In simple terms, the chain rule provides a way to differentiate composite functions.For the polar equation given, the process is:
  • Recognize that \( a\sin\theta \) involves a multiplication of a constant \( a \) with \( \sin\theta \), which is our inner function.
  • The derivative of \( \sin\theta \) is \( \cos\theta \).
  • Apply the chain rule: differentiate \( a\sin\theta \), yielding \( \frac{dr}{d\theta} = a\cos\theta \).
This calculation is crucial when substituting values into the arc length formula, as it directly affects the simplicity and accuracy of the integration process that follows.
Derivatives
In the context of polar curves, derivatives play a vital role in evaluating the changing rates along the curve. Here, we compute the derivative of the function \( r = a\sin\theta \) to gain insight into the curve's orientation and velocity at different points.A derivative, in this setting, reveals:
  • The relationship between changes in \( r \) and \( \theta \).
  • Identifies the maxima and minima points of the curve, indicative of directional shifts.
For the exercise, we've found that \( \frac{dr}{d\theta} = a\cos\theta \), which means the impact of the angle \( \theta \) on the radial distance is modulated by the cosine function. Such computations aid in tackling integration tasks where knowing how the curve's radius varies aids in precise length calculations.
Integration Bounds
The bounds of integration are essential in defining the section of a curve you're interested in measuring or analyzing. For entire curves like complete circles in polar coordinates, these bounds encompass a full cycle.Consider:
  • For a complete circle represented by \( r = a\sin\theta \), the bounds are from \( 0 \) to \( 2\pi \).
  • These bounds ensure that the integration covers the entire curve, from its inception back to the initial point.
Correctly identifying these bounds is critical in ensuring that calculations, particularly those involving length, are accurate and reflect the complete geometry intended by the curve's equation. In practice, this means setting the integration limts as \( \alpha = 0 \) and \( \beta = 2\pi \) to capture every segment of the polar path.

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Most popular questions from this chapter

Assume \(m\) is a positive integer. a. Even number of leaves: What is the relationship between the total area enclosed by the \(4 m\) -leaf rose \(r=\cos (2 m \theta)\) and \(m ?\) b. Odd number of leaves: What is the relationship between the total area enclosed by the \((2 m+1)\) -leaf rose \(r=\cos ((2 m+1) \theta)\) and \(m ?\)

Which of the following parametric equations describe the same curve? a. \(x=2 t^{2}, y=4+t ;-4 \leq t \leq 4\) b. \(x=2 t^{4}, y=4+t^{2} ;-2 \leq t \leq 2\) c. \(x=2 t^{2 / 3}, y=4+t^{1 / 3} ;-64 \leq t \leq 64\)

Consider a hyperbola to be the set of points in a plane whose distances from two fixed points have a constant difference of \(2 a\) or \(-2 a\). Derive the equation of a hyperbola. Assume the two fixed points are on the \(x\) -axis equidistant from the origin.

Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of \(t\). $$x=e^{t}, y=\ln (t+1) ; t=0$$

Find the length of the following polar curves. The spiral \(r=2 e^{2 \theta},\) for \(0 \leq \theta \leq \ln 8\)
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