4.3.E: Line Integrals (Exercises)

Exercise \(\PageIndex{1}\)

For each of the following, compute the line integral \(\int_{C} F \cdot d s\) for the given vector field \(F\) and curve \(C\) parametrized by \(\varphi\).

(a) \(F(x, y)=(x y, 3 x), \varphi(t)=\left(t^{2}, t\right), 0 \leq t \leq 2\)

(b) \(F(x, y)=\left(\frac{y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right), \varphi(t)=(\cos (t), \sin (t)), 0 \leq t \leq 2 \pi\)

(c) \(F(x, y)=\left(3 x-2 y, 4 x^{2} y\right), \varphi(t)=\left(t^{3}, t^{2}\right),-2 \leq t \leq 2\)

(d) \(F(x, y, z)=\left(x y z, 3 x y^{2}, 4 z\right), \varphi(t)=\left(3 t, t^{2}, 4 t^{3}\right), 0 \leq t \leq 4\)

Answer

(a) \(\int_{C} F \cdot d s=\frac{104}{5}\)

(c) \(\int_{C} F \cdot d s=-\frac{384}{5}\)

Exercise \(\PageIndex{2}\)

Let \(C\) be the circle of radius 2 centered at the origin in \(\mathbb{R}^2\), with counterclockwise orientation. Evaluate the following line integrals.

(a) \(\int_{C} 3 x d x+4 y d y\)

(b) \(\int_{C} 8 x y d x+4 x^{2} d y\)

Answer

(a) \(\int_{C} 3 x d x+4 y d y=0\)

Exercise \(\PageIndex{3}\)

Let \(C\) be the part of a helix in \(\mathbb{R}^3\) parametrized by \(\varphi(t)=(\cos (2 t), \sin (2 t), t), 0 \leq t \leq 2\pi \). Evaluate the following line integrals.

(a) \(\int_{C} 3 x d x+4 y d y+z d z\)

(b) \(\int_{C} y z d x+x z d y+x y d z\)

Answer

(a) \(\int_{C} 3 x d x+4 y d y+z d z=2 \pi^{2}\)

Exercise \(\PageIndex{4}\)

Let \(C\) be the rectangle in \(\mathbb{R}^2\) with vertices at (−1,1), (2,1), (2,3), and (−1,3), with counterclockwise orientation. Evaluate the following line integrals.

(a) \(\int_{C} x^{2} y d x+(3 y+x) d y\)

(b) \(\int_{C} 2 x y d x+x^{2} d y\)

Answer

(a) \(\int_{C} x^{2} y d x+(3 y+x) d y=0\)

Exercise \(\PageIndex{5}\)

Let \(C\) be the ellipse in \(\mathbb{R}^2\) with equation

\[ \frac{x^{2}}{4}+\frac{y^{2}}{9}=1 , \nonumber \]

with counterclockwise orientation. Evaluate \(\int_{C} F \cdot d s \text { for } F(x, y)=(4 y, 3 x)\).

Answer

\(\int_{C} F \cdot d s=-6 \pi\)

Exercise \(\PageIndex{6}\)

Let \(C\) be the upper half of the circle of radius 3 centered at the origin in \(\mathbb{R}^2\), with counterclockwise orientation. Evaluate the following line integrals.

(a) \(\int_{C} 3 y d x\)

(b) \(\int_{C} 4 x d y\)

Answer

(a) \(\int_{C} 3 y d x=-\frac{27 \pi}{2}\)

Exercise \(\PageIndex{7}\)

Evaluate

\[ \int_{C} \frac{x}{x^{2}+y^{2}} d x+\frac{y}{x^{2}+y^{2}} d y , \nonumber \]

where \(C\) is any curve which starts at (1,0) and ends at (2,3).

Answer

\(\int_{C} \frac{x}{x^{2}+y^{2}} d x+\frac{y}{x^{2}+y^{2}} d y=\frac{1}{2} \log (13)\)

Exercise \(\PageIndex{8}\)

(a) Suppose \(F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is a \(C^1\) vector field which is the gradient of a scalar function \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\). If \(F_k\) is the \(k\)th coordinate function of \(F, k=1,2, \ldots, n\), show that

\[ \frac{\partial}{\partial x_{j}} F_{i}\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{\partial}{\partial x_{i}} F_{j}\left(x_{1}, x_{2}, \ldots, x_{n}\right) \nonumber \]

for \(i=1,2, \ldots, n\) and \(j=1,2, \ldots, n\).

(b) Show that although

\[ \int_{C} x d x+x y d y=0 \nonumber \]

for every circle \(C\) in \(\mathbb{R}^2\) with center at the origin, nevertheless \(F(x, y)=(x, x y)\) is not the gradient of any scalar function \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\).

(c) Let

\[ F(x, y)=\left(-\frac{y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right) \nonumber \]

for all \((x,y)\) in the set \(S=\{(x, y):(x, y) \neq(0,0)\}\). Let \(F_1\) and \(F_2\) be the coordinate functions of \(F\). Show that

\[ \frac{\partial}{\partial y} F_{1}(x, y)=\frac{\partial}{\partial x} F_{2}(x, y) \nonumber \]

for all \((x,y)\) in \(S\), even though \(F\) is not the gradient of any scalar function. (Hint: For the last part, show that

\[ \int_{C} F \cdot d s=2 \pi , \nonumber \]

where \(C\) is the unit circle centered at the origin.)

Exercise \(\PageIndex{9}\)

Suppose \(F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is a continuous vector field with the property that for any curve \(C\),

\[ \int_{C} F \cdot d s \nonumber \]

depends only on the endpoints of \(C\). That is, if \(C_1\) and \(C_2\) are any two curves with the same endpoints \(P\) and \(Q\), then

\[ \int_{C_{1}} F \cdot d s=\int_{C_{2}} F \cdot d s . \nonumber \]

(a) Show that

\[ \int_{C} F \cdot d s=0 \nonumber \]

for any closed curve \(C\).

(b) Let \(F_1\) and \(F_2\) be the coordinate functions of \(F\). Define \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) by

\[ f(x, y)=\int_{C} F \cdot d s , \nonumber \]

where \(C\) is any curve which starts at (0,0) and ends at \((x,y)\). Show that

\[ \frac{\partial}{\partial y} f(x, y)=F_{2}(x, y) . \nonumber \]

(Hint: In evaluating \(f(x,y)\), consider the curve \(C\) from (0,0) to \((x,y)\) which consists of the horizontal line from (0,0) to (\(x\),0) followed by the vertical line from (\(x\),0) to (\(x\),\(y\)).)

(c) Show that \(\nabla f=F\).