5.4 Using Parameterizations to Calculate Line Integrals

Using Parameterizations to Calculate Line Integrals

Introduction

Parameterizations in the Definition of

Alternative Notation for Line Integrals

Let $\vF=\langle xy,y^2\rangle$, let $C_1$ be the line segment from $(1,1)$ to $(4,1)$, let $C_2$ be the line segment from $(4,1)$ to $(4,3)$, and let $C_3$ be the line segment from $(1,1)$ to $(4,3)$. Also let $C = C_1 + C_2$. This vector field and the curves are shown in .

Every point along $C_1$ has $y=1$. Therefore, along $C_1$, the vector field $\vF$ can be viewed purely as a function of $x$. In particular, along $C_1$, we have $\vF(x,1) = \langle x,1\rangle$. Since every point along $C_2$ has the same $x$-value, write $\vF$ as a function of $y$ only (for the points on $C_2$).

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

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

Let $\vF(x,y) = x\vi + y^2\vj$ and let $C$ be the quarter of the circle of radius $3$ from $(0,3)$ to $(3,0)$. This vector field and curve are shown in . By properties of line integrals, we know that $\int_C \vF\cdot d\vr = -\int_{-C}\vF\cdot d\vr$, and we will use this property since $-C$ is the usual clockwise orientation of a circle, meaning we can parametrize $-C$ by $\vr(t) = \langle 3\cos(t),3\sin(t)\rangle$ for $0\leq t\leq \pi/2$.

To evaluate $\int_{-C}\vF\cdot d\vr$ using this parametrization, we need to note that

Thus, we have

. Note that we have used the substitution $u =\sin(t)$ in evaluating the definite integral here.

Find the work done by the vector field $\vF(x,y,z) = 6x^2z\vi + 3y^2\vj + x\vk$ on a particle that moves from the point $(3,0,0)$ to the point $(3,0,6\pi)$ along the helix given by $\vr(t) = \langle 3\cos(t),3\sin(t),t\rangle$.

Let $\vF(x,y) = \langle 0,x\rangle$. Let $C$ be the closed curve consisting of the top half of the circle of radius $2$ centered at the origin and the portion of the $x$-axis from $(2,0)$ to $(-2,0)$, oriented clockwise. Find the circulation of $\vF$ around $C$.

Let $C_1$ be the portion of the graph of $y=2x^3+3x^2-12x-15$ from $(-2,5)$ to $(3,30)$. Calculate $\int_{C_1}\vF\cdot d\vr$.

Let $C_2$ be the line segment from $(-2,5)$ to $(3,30)$. Calculate $\int_{C_2}\vF\cdot d\vr$.

Let $C_3$ be the circle of radius $3$ centered at the origin, oriented counterclockwise. Calculate $\oint_{C_3} \vF\cdot d\vr$.

Let $\vF(x,y) = \langle y^2,2xy+3\rangle$.