/*! 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 13 a. Set up an integral for the le... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 13

a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=x^{2}, \quad-1 \leq x \leq 2$$

Short Answer

Expert verified
The length of the curve is approximately 5.386.

Step by step solution

01

Formula for Arc Length

The formula for the length of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is given by \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]Here, \( f(x) = x^2 \), so the derivative, \( \frac{dy}{dx} = 2x \).
02

Set Up the Integral

Substitute \( \frac{dy}{dx} = 2x \) into the arc length formula:\[ L = \int_{-1}^{2} \sqrt{1 + (2x)^2} \, dx = \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \]
03

Graph the Curve

Using graphing software or a calculator, plot the curve \( y = x^2 \) over the interval \(-1 \leq x \leq 2\). The curve is a portion of a parabola opening upwards.
04

Evaluate the Integral Numerically

Input the integral \[ \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \]into a calculator or computer software that can evaluate integrals numerically. The numerical solution for the length of the curve is approximately 5.386.

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.

Arc Length Formula
Finding the arc length of a curve is a common task in calculus. The arc length formula is essential for this calculation. It helps in determining the length of a smooth curve over a specific interval. The general formula for the arc length of the curve described by the function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by the integral: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
  • \( L \) is the length of the arc.
  • \( \frac{dy}{dx} \) is the derivative of the function \( f(x) \).
  • This formula accounts for both the vertical and horizontal changes along the curve, providing the total distance along the curve.
By using this formula, you can compute the arc length by integrating over the defined interval, ensuring you consider the curve's shape and slope.
Derivative
The derivative is a fundamental building block in calculus, representing how a function changes as its input changes. For the arc length calculation, we need to know how steep or flat a curve is at any point. This means finding the derivative, \( \frac{dy}{dx} \), of the function. For example, if \( y = x^2 \), its derivative is \( \frac{dy}{dx} = 2x \).
  • The derivative helps determine the slope at any point on the curve.
  • For arc length calculations, you'll plug this derivative into the formula for arc length.
  • It provides an understanding of the curve's steepness, which is crucial in measuring the distance along the curve.
Understanding derivatives not only assists in calculating arc lengths but also in analyzing the behavior and properties of curves.
Numerical Integration
Sometimes, computing integrals analytically (i.e., finding an exact answer using algebra) is either too complicated or impossible. That's where numerical integration comes in handy. It approximates the value of an integral using numerical methods, which essentially means using calculations to get as close as possible to the real value. For our exercise, the integral \[ \int_{-1}^{2} \sqrt{1 + 4x^2} \, dx \] can be evaluated using schemes such as Simpson's Rule or the Trapezoidal Rule. Today, software and calculators can handle this easily:
  • Numerical integration provides an approximate value, useful when an exact solution is hard to find.
  • It's implemented in most graphing calculators and computer software like Python, MATLAB, or specific integral solvers you might use.
  • This technique enables us to solve complex problems in physics, engineering, and other sciences where continuous functions describe real-world phenomena.
Graphing Curves
Graphing curves is a valuable skill to visually understand the behavior of equations and mathematical functions. When you graph the function \( y = x^2 \), you'll see a parabola that opens upwards, providing a visual representation of its arc length, slope, and curvature.
  • Allows you to see how a function behaves over a certain interval.
  • Visual aids such as graphs can make it easier to comprehend abstract mathematical concepts.
  • Graphing tools can also directly compute properties such as intercepts, turning points, and symmetries of a curve.
For example, by graphing \( y=x^2 \) over \(-1 \leq x \leq 2\), you'll get a sense of the curve's geometry, further aiding in confirming results from analytical calculations or numerical integrations.

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

For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the \(y\) -axis, for example, and washers are used, we must integrate with respect to \(y .\) It may not be possible, however, to express the integrand in terms of \(y .\) In such a case, the shell method allows us to integrate with respect to \(x\) instead. Compute the volume of the solid generated by revolving the triangular region bounded by the lines \(2 y=x+4, y=x,\) and \(x=0\) about a. the \(x\) -axis using the washer method. b. the \(y\) -axis using the shell method. c. the line \(x=4\) using the shell method. d. the line \(y=8\) using the washer method.

Derive the formula for the volume of a right circular cone of height \(h\) and radius \(r\) using an appropriate solid of revolution.

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines. \(y=x+2, \quad y=x^{2}\) a. The line \(x=2\) b. The line \(x=-1\) c. The \(x\) -axis d. The line \(y=4\)

Volume of a bowl a. A hemispherical bowl of radius \(a\) contains water to a depth \(h\) Find the volume of water in the bowl. b. Related rates Water runs into a sunken concrete hemispherical bowl of radius \(5 \mathrm{m}\) at the rate of \(0.2 \mathrm{m}^{3} / \mathrm{s}\). How fast is the water level in the bowl rising when the water is \(4 \mathrm{m}\) deep?

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. \(x=\left(y^{4} / 4\right)+1 /\left(8 y^{2}\right), \quad 1 \leq y \leq 2 ; \quad x\) -axis \(\quad\) (Hint: \(\quad\) Express \(d s=\sqrt{d x^{2}+d y^{2}}\) in terms of \(d y,\) and evaluate the integral \(S=\int 2 \pi y d s\) with appropriate limits.)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.