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Chapter 6: Problem 45

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. $$ [\mathrm{T}] \int_{C}(x-y) d s $$ $$ C : \mathbf{r}(t)=4 t \mathbf{i}+3 t \mathbf{j} \text { when } 0 \leq t \leq 2 $$

Short Answer

Expert verified
The value of the line integral is 10.

Step by step solution

01

Understand the Problem

We need to evaluate the line integral \( \int_{C}(x-y) \, ds \) where the curve \( C \) is given by the parametric equation \( \mathbf{r}(t) = 4t \mathbf{i} + 3t \mathbf{j} \) for \( t \) in the interval \( [0, 2] \). The expression \( ds \) represents the differential arc length along the curve.
02

Find Parametric Derivatives

Calculate the derivatives of the parametric equations: \( \frac{dx}{dt} = \frac{d}{dt}(4t) = 4 \) and \( \frac{dy}{dt} = \frac{d}{dt}(3t) = 3 \). These will be used to find \( ds \).
03

Calculate Differential Arc Length \( ds \)

The differential arc length \( ds \) is given by \( ds = \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \). Substitute the derivatives to get \( ds = \sqrt{4^2 + 3^2} \, dt = 5 \, dt \).
04

Substitute into Integral Expression

Substitute \( x = 4t \) and \( y = 3t \) into the integral: \( \int_{C}(x-y) \, ds = \int_{0}^{2} ((4t) - (3t)) \, 5 \, dt = \int_{0}^{2} 5t \, dt \).
05

Evaluate the Integral

Integrate \( \int 5t \, dt \) over the interval from \( 0 \) to \( 2 \): \[ \int_{0}^{2} 5t \, dt = 5 \left[ \frac{t^2}{2} \right]_{0}^{2} = 5 \left( \frac{4}{2} - 0 \right) = 5 \times 2 = 10 \].
06

State the Final Result

The result of the line integral is \( 10 \).

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Key Concepts

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

Parametric Equations
Parametric equations are a way to define a curve using one or more parameters, such as \( t \), instead of using a standard function format like \( y = f(x) \). In this exercise, the curve \( C \) is given by the parametric equations \( \mathbf{r}(t) = 4t \mathbf{i} + 3t \mathbf{j} \). Here, \( t \) is the parameter that varies from 0 to 2.

  • Component Form: The expression \( \mathbf{r}(t) = 4t \mathbf{i} + 3t \mathbf{j} \) indicates that as \( t \) changes, the curve traces a path on a plane.
  • Coordinates: The functions for \( x \) and \( y \) are \( x(t) = 4t \) and \( y(t) = 3t \), representing the x-coordinate and y-coordinate, respectively.
These parametric equations help in expressing the geometry and properties of a curve, making it easier to solve certain types of calculus problems, such as evaluating line integrals.
Differential Arc Length
The concept of differential arc length involves finding the length of a small segment of a curve, denoted as \( ds \). It is crucial for calculating line integrals over a curve. In this case, the differential arc length \( ds \) is derived from the derivatives of the parametric equations.

  • Formula: \( ds = \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \).
  • Calculation: By substituting \( \frac{dx}{dt} = 4 \) and \( \frac{dy}{dt} = 3 \), we find \( ds = \sqrt{4^2 + 3^2} \, dt = 5 \, dt \).
The differential arc length consolidates the shape of the curve into a calculable form, allowing for the integration over its path.
Computer Algebra System (CAS)
A Computer Algebra System (CAS) is a software tool that facilitates symbolic mathematics operations, including the evaluation of line integrals such as the one in this exercise. With a CAS, computations are more manageable, especially when working with complex expressions or equations.

  • Simplification: CAS automates and simplifies many steps in mathematical calculations, like differentiating functions or transforming integral expressions.
  • Accuracy: It provides accurate and efficient solutions, reducing the possibility of human error during lengthy calculations.
  • Application: In this exercise, a CAS could be used to automatically compute the integral \( \int_{0}^{2} 5t \, dt \) to yield the final result of 10.
By leveraging a CAS, students can focus on understanding the underlying mathematical concepts rather than getting bogged down in arithmetic.

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Most popular questions from this chapter

For the following exercises, find the work done. Find the work done by force \(\mathbf{F}(x, y)=2 y \mathbf{i}+3 x \mathbf{j}+(x+y) \mathbf{k} \quad\) in moving an object along curve \(\mathbf{r}(t)=\cos (t) \mathbf{i}+\sin (t) \mathbf{j}+\frac{1}{6} \mathbf{k}, \quad\) where \(0 \leq t \leq 2 \pi\)

For the following exercises, find the flux. Let \(\mathbf{F}=5 \mathbf{j}\) and let \(C\) be curve \(y=0,0 \leq x \leq 4\) Find the flux across \(C .\)

For the following exercises, evaluate the line integrals. Evaluate \(\int_{C} y z d x+x z d y+x y d z\) over the line segment from \((1,1,1)\) to \((3,2,0)\)

In the following exercises, find the work done by force field \(\mathbf{F}\) on an object moving along the indicated path. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}-5 z \mathbf{k}\) \(C : \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k}, 0 \leq t \leq 2 \pi\)

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate line integral \(\int_{\gamma} x y^{4} d s,\) where \(\gamma\) is the right half of circle \(x^{2}+y^{2}=16\) .
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