/*! 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 61 Give the integral formulas for a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 61

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

Short Answer

Expert verified
The formula for area in polar coordinates is \( A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta \). The formula for arc length in polar coordinates is \( L = \int_{a}^{b} \sqrt{[f(\theta)]^2 + [f'(\theta)]^2} d\theta \).

Step by step solution

01

Formula for Area in Polar Coordinates

In polar coordinates, if a region is defined by \(\theta = a\) and \(\theta = b\), and a curve \(r = f(\theta)\) where \(a \le \theta \le b\), the area A of the region is given by the integral formula: \[ A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta \] This formula gives the area of a sector defined by angles a and b and a curve defined by the polar equation r.
02

Formula for Arc Length in Polar Coordinates

The arc length of a curve in polar coordinates from \(\theta = a\) to \(\theta = b\), given by the equation \(r = f(\theta)\), can be calculated using the integral formula: \[ L = \int_{a}^{b} \sqrt{[f(\theta)]^2 + [f'(\theta)]^2} d\theta \] This formula takes into account both the radial and angular changes contributing to the length of the curve.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integral Formulas
Integral formulas play a crucial role in calculus, especially when dealing with polar coordinates. Unlike Cartesian coordinates, polar coordinates describe a point in a plane using a radius and an angle. This unique representation requires different approaches when calculating areas and lengths.

These formulas help us integrate and obtain values by summing infinitely small elements to find totals over a specific range. They form the basis for finding areas and arc lengths when working with curves defined in polar systems. With these tools, we can explore complex shapes and curves in various fields such as physics, engineering, and more.
Area in Polar Coordinates
Finding the area of a region in polar coordinates involves a special integral formula. When a curve is defined as \(r = f(\theta)\), the area \(A\) between angles \(a\) and \(b\) is given by:

\[ A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta \]

This formula essentially sums up tiny wedges of area spanning between the two angles.
  • Each wedge has a small angle \(d\theta\).
  • The squared function \([f(\theta)]^2\) accounts for the distance from the origin to the curve at each angle.
By integrating these elements, you accumulate the entire area's value contained between the specified bounds.
Arc Length in Polar Coordinates
Calculating the arc length of a curve in polar coordinates requires a bit more complexity. The arc length \(L\) from \(\theta = a\) to \(\theta = b\) for a curve \(r = f(\theta)\) is calculated by:

\[ L = \int_{a}^{b} \sqrt{[f(\theta)]^2 + [f'(\theta)]^2} d\theta \]

This formula combines both radial and angular components:
  • \([f(\theta)]^2\) provides the radial contribution.
  • \([f'(\theta)]^2\) captures the change in radius as the angle changes.
The square root ensures the total contribution aligns with the distance, giving you the accurate length of the curve.
Calculus
Calculus is the mathematical framework that allows us to work with concepts of change and accumulation such as those seen in polar coordinates. It includes differentiation and integration, providing tools to analyze motion, growth patterns, and changes within systems.

In the world of polar coordinates, calculus helps bridge the gap between complex shapes and numerical analysis. By using integral formulas, we can capture essential physical attributes such as areas and lengths in a precise and efficient manner. Understanding calculus in polar contexts opens doors to deeper insights into natural patterns and technological applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the area of the surface generated by revolving the curve about each given axis. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Interval }} \\\ x=4 \cos \theta, y=4 \sin \theta, &\quad 0 \leq \theta \leq \frac{\pi}{2}, \quad y \text { -axis } \end{array} $$

Show that the graphs of the equations intersect at right angles: \(\frac{x^{2}}{a^{2}}+\frac{2 y^{2}}{b^{2}}=1 \quad\) and \(\quad \frac{x^{2}}{a^{2}-b^{2}}-\frac{2 y^{2}}{b^{2}}=1\)

(a) Each set of parametric equations represents the motion of a particle. Use a graphing utility to graph each set. $$ \begin{aligned} &\begin{array}{ll} \underline{\text { First Particle }} \\ x=3 \cos t \end{array} \quad \frac{\text { Second Particle }}{x=4 \sin t}\\\ &\begin{array}{ll} y=4 \sin t && y=3 \cos t \\ 0 \leq t \leq 2 \pi && 0 \leq t \leq 2 \pi \end{array} \end{aligned} $$

Sketch a graph of the polar equation. $$ r=4(1+\cos \theta) $$

Find the arc length of the curve on the interval \([0,2 \pi]\). Hypocycloid perimeter: \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.