Line Integrals

If the equation of a curve is given either as a single equation $f(x,y)=0$, or parametrically: $x=\varphi (t)$, $y=\psi (t)$, then the line integral along the curve $AB$, denoted by $\displaystyle \int_{AB} P(x,y) \, dx+Q(x,y) \, dy$, is evaluated by reducing it to an ordinary definite integral: if $x=\varphi (t)$, $y=\psi (t)$ and the parameter $t$ varies from $a$ to $b$ as we pass from $A$ to $B$, then

When the direction of the path of integration is reversed, the value of the integral is multiplied by minus one: $\displaystyle \int_{AB} P \, dx+Q \, dy=-\int_{BA} P \, dx+Q \, dy$.

Find the value of the integral $\displaystyle \int_{AB} x \, dy-y \, dx$ along the parabola $y=x^{2}$ between the points $A(0,0)$ and $B(2,4)$.

Find the integral $\displaystyle \int_{AB}(x+y) \, dx+(x-y) \, dy$, if the path of integration is the parabola $y=x^{2}$ between the points $(-1,1)$, $(1,1)$.

Find the integral $\displaystyle \int_{AB} x^{2} \, ds$, where the path of integration is the upper half of the circle $x^{2}+y^{2}=a^{2}$ between the points $A(a,0)$ and $B(-a,0)$.

Find the value of the integral $\displaystyle \int_{AB} x \, dy$, taken along the right half-circle $x^{2}+y^{2}=a^{2}$ between the points $A(0,-a)$ and $B(0,a)$.

Find the value of the integral $\displaystyle \int_{AB}(x^{2}+y^{2}) \, dx+(x^{2}-y^{2}) \, dy$, taken around the contour of the triangle with vertices at the points $A(0,0)$, $B(1,0)$, $C(0,1)$.

From Green's formula (see problem 3621) derive the equality: $S=\dfrac {1}{2}\int_{C}(x \, dy-y \, dx)$, where $C$ is a contour enclosing the area $S$ in the positive direction.

Using the formula of problem 3622 ($S=\dfrac {1}{2}\int_{C}(x\, dy-y\, dx)$), compute the area of the ellipse $x=a\cos t$, $y=b\sin t$.

Using the formula of problem 3622 ($S=\dfrac {1}{2}\int_{C}(x\, dy-y\, dx)$), compute the area of the sector of the hyperbola $x=a\operatorname {ch}t$, $y=b\operatorname {sh}t$, bounded by the axis $Ox$, the arc of the hyperbola up to the point $M(x_{0},y_{0})$, and the line $OM$.

Using the formula of problem 3622 ($S=\dfrac {1}{2}\int_{C}(x\, dy-y\, dx)$), compute the area of the sector of the astroid $x=a\cos^{3}t$, $y=a\sin^{3}t$, bounded by the lines $OM_{1}$ and $OM_{2}$ from the origin and the arc $M_{1}M_{2}$ of the astroid (the points $M_{1}$, $M_{2}$ correspond to the parameter values $t_{1}$, $t_{2}$).

Find the area of the three-cusped hypocycloid:

Find the value of the line integral $\displaystyle \int_{C} \dfrac {x \, dy-y \, dx}{x^{2}+y^{2}}$, taken around a closed contour $C$ in the positive direction.

The curves $X(x,y)=0$ and $Y(x,y)=0$ have several simple points of intersection. Compute the value of the integral $\dfrac {1}{2\pi }\displaystyle \int_{C} \dfrac {X\, dY-Y\, dX}{X^{2}+Y^{2}}$, taken in the positive direction around a closed contour $C$.

One application of line integrals is finding the center of gravity $(x_{c}, y_{c}, z_{c})$ of a curve. For this the following formulas are used:

where $\mu (x,y,z)$ is the density of the curve at the point $(x,y,z)$, and $M$ is the mass of the curve.

Find the coordinates of the center of gravity of a homogeneous arc of a circle of radius $a$ subtending a central angle $2\varphi$.

Find the coordinates of the center of gravity of the entire arc of the homogeneous cardioid $r=a(1+\cos \varphi )$.

Find the coordinates of the center of gravity of one arch of the homogeneous cycloid

Find the coordinates of the center of gravity of an arc of the helix

for $0<t<m$.

Find the coordinates of the center of gravity of a homogeneous arc of the curve

for $0<t<\infty$.