5.3 The Idea of a Line Integral

The Idea of a Line Integral

Introduction

Orientations of Curves

Line Integrals

Properties of Line Integrals

The Circulation of a Vector Field

Recall from Section 1.4 that the work done by a force $\vF$ on an object that moves with displacement vector $\vv$ is $\vF\cdot \vv$. In this Preview Activity, we consider the work done by wind on a helicopter at various stages of its journey.

Our intrepid pilot flies for some time and finds that they are 30 from where they started at a heading of 20 degrees east of due north. During this portion of the trip, the wind is exerting a force of 100 on the helicopter in the due east direction. Find the work the wind has done on the helicopter during the flight.

The line segment in $\R^3$ from $(0,1,-2)$ to $(3,-1,2)$.

The line segment in $\R^3$ from $(3,-1,2)$ to $(0,1,-2)$.

The circle of radius $3$ (in $\R^2$) centered at the origin, beginning at the point $(0,-3)$ and proceeding clockwise around the circle.

In $\R^2$, the portion of the parabola $y^2 = x$ from the point $(4,2)$ to the point $(1,-1)$.

$\displaystyle\int_{C_1}\vF\cdot d\vr$

$\displaystyle\int_{C_2}\vF\cdot d\vr$

$\displaystyle\int_{C_3}\vG\cdot d\vr$

$\displaystyle\int_{C_4}\vG\cdot d\vr$

Work through this exercise and explain your reasoning step by step.

Is $\int_{C_6}\vF\cdot d\vr$ positive, negative, or zero? Explain.

Let $C = C_1+C_2+C_3+C_4$. Determine if $\displaystyle\int_C\vF\cdot d\vr$ is positive, negative, or zero.

Order the line integrals below from smallest to largest.

Let $C$ be the path given below from $P$ to $Q$ with pieces $C_1$, $C_2$, and $C_3$ as labeled. Let $\vF$ be a vector field such that $\int_C \vF\cdot d\vr = 9$, $\int_{C_1} \vF\cdot d\vr = 6$,and $\int_{C_3} \vF\cdot d\vr = 7$.

Find the following: -$\int_{-C_3} \vec{\vF}\cdot d\vec{\vr}$-$\int_{C_2} \vec{\vF}\cdot d\vec{\vr}$-$\int_{-C_1-C_3} \vec{\vF}\cdot d\vec{\vr}$