Curve Integrals

Compute the following curve integral of the first type: $\displaystyle \int_{C} (x+y)\, ds$, where $C$ is the triangular contour with vertices $O(0,0)$, $A(1,0)$, and $B(0,1)$.

Compute $\displaystyle \int_{C} y^{2}\, ds$, where $C$ is one arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)\ (0\leqslant t\leqslant 2\pi )$.

Compute $\displaystyle \int_{C} (x^{2}+y^{2})\, ds$, where $C$ is the curve $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)\ (0\leqslant t\leqslant 2\pi )$.

Compute $\displaystyle \int_{C} xy\, ds$, where $C$ is the arc of the hyperbola $x=a\operatorname {ch} t$, $y=a\operatorname {sh} t\ (0\leqslant t\leqslant t_{0})$.

Compute $\displaystyle \int_{C} \left(x^{4/3}+y^{4/3}\right)\, ds$, where $C$ is the arc of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$.

Compute $\displaystyle \int_{C} e^{\sqrt{x^{2}+y^{2}}}\, ds$, where $C$ is the convex contour bounded by the curves $r=a$, $\varphi =0$, $\varphi =\dfrac {\pi }{4}$ ($r$ and $\varphi$ being polar coordinates).

Compute $\displaystyle \int_{C} \left|y\right|\, ds$, where $C$ is the arc of the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$.

Compute $\displaystyle \int_{C} x\, ds$, where $C$ is the part of the logarithmic spiral $r=ae^{k\varphi }\ (k>0)$ inside the circle $r=a$.

Compute $\displaystyle \int_{C} \sqrt{x^{2}+y^{2}}\, ds$, where $C$ is the circle $x^{2}+y^{2}=ax$.

Compute $\displaystyle \int_{C} \dfrac {ds}{y^{2}}$, where $C$ is the catenary $y=a\operatorname {ch}\dfrac {x}{a}$.

Find the arc length of the following space curve (parameter taken positive): $x=3t$, $y=3t^{2}$, $z=2t^{3}$, from $O(0,0,0)$ to $A(3,3,2)$.

Find the arc length of the following space curve (parameter taken positive): $x=e^{-t}\cos t$, $y=e^{-t}\sin t$, $z=e^{-t}$, for $0\leqslant t<+\infty$.

Find the arc length of the following space curve (parameter taken positive): $y=a\arcsin \dfrac {x}{a}$, $z=\dfrac {a}{4}\ln \dfrac {a-x}{a+x}$, from $O(0,0,0)$ to $A(x_{0},y_{0},z_{0})$.

Find the arc length of the following space curve (parameter taken positive): $(x-y)^{2}=a(x+y)$, $x^{2}-y^{2}=\dfrac {9}{8}z^{2}$, from $O(0,0,0)$ to $A(x_{0},y_{0},z_{0})$.

Find the arc length of the following space curve (parameter taken positive): $x^{2}+y^{2}=cz$, $\dfrac {y}{x}=\operatorname {tg}\dfrac {z}{c}$, from $O(0,0,0)$ to $A(x_{0},y_{0},z_{0})$.

Find the arc length of the following space curve (parameter taken positive): $x^{2}+y^{2}+z^{2}=a^{2}$, $\sqrt{x^{2}+y^{2}}\operatorname {ch}\left(\operatorname {arctg}\dfrac {y}{x}\right)=a$, from point $A(a,0,0)$ to point $B(x,y,z)$.

Compute the following first-type curve integral taken along a space curve: $\displaystyle \int_{C} (x^{2}+y^{2}+z^{2})\, ds$, where $C$ is a segment of the helix $x=a\cos t$, $y=a\sin t$, $z=bt\ (0\leqslant t\leqslant 2\pi )$.

Compute the following first-type curve integral taken along a space curve: $\displaystyle \int_{C} x^{2}\, ds$, where $C$ is the circle $x^{2}+y^{2}+z^{2}=a^{2}$, $x+y+z=0$.

Compute the following first-type curve integral taken along a space curve: $\displaystyle \int_{C} z\, ds$, where $C$ is the conical spiral $x=t\cos t$, $y=t\sin t$, $z=t\ (0\leqslant t\leqslant t_{0})$.

Compute the following first-type curve integral taken along a space curve: $\displaystyle \int_{C} z\, ds$, where $C$ is the arc of the curve $x^{2}+y^{2}=z^{2}$, $y^{2}=ax$ from the point $O(0,0,0)$ to the point $A(a,a,a\sqrt{2})$.

Suppose the curve $x=a\cos t$, $y=b\sin t\ (0\leqslant t\leqslant 2\pi )$ has linear density $\rho =\left|y\right|$ at the point $(x,y)$; find its mass.

Find the mass of the arc of the curve $x=at$, $y=\dfrac {a}{2}t^{2}$, $z=\dfrac {a}{3}t^{3}\ (0\leqslant t\leqslant 1)$, whose density varies according to the law $\rho =\sqrt{\dfrac {2y}{a}}$.

Compute the coordinates of the centroid of the arc of the uniform curve $y=a\operatorname {ch}\dfrac {x}{a}$ from the point $A(0,a)$ to the point $B(b,h)$.

Find the centroid of the arc of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)\ (0\leqslant t\leqslant \pi )$.

Compute the coordinates of the centroid of the boundary of the spherical triangle $x^{2}+y^{2}+z^{2}=a^{2}$; $x\geqslant 0$, $y\geqslant 0$, $z\geqslant 0$.

Find the coordinates of the centroid of the uniform arc $x=e^{t}\cos t$, $y=e^{t}\sin t$, $z=e^{t}\ (-\infty <t\leqslant 0)$.

Find the moment of inertia, about the coordinate axes, of one turn of the helix $x=a\cos t$, $y=a\sin t$, $z=\dfrac {h}{2\pi }t\ (0\leqslant t\leqslant 2\pi )$.

Compute the second-type curve integral $\displaystyle \int_{OA} x\, dy-y\, dx$, where $O$ is the origin of coordinates, the coordinates of point $A$ are $(1,2)$, and:

For the paths indicated in problem 4248, compute $\displaystyle \int_{OA} x\, dy+y\, dx$, where $O$ is the origin of coordinates, the coordinates of point $A$ are $(1,2)$, and:

In the direction of increasing parameter, along the indicated curve compute the following second-type curve integral: $\displaystyle \int_{C} (x^{2}-2xy)\, dx+(y^{2}-2xy)\, dy$, where $C$ is the parabola $y=x^{2}\ (-1\leqslant x\leqslant 1)$.

In the direction of increasing parameter, along the indicated curve compute the following second-type curve integral: $\displaystyle \int_{C} (x^{2}+y^{2})\, dx+(x^{2}-y^{2})\, dy$, where $C$ is the curve $y=1-\left|1-x\right|\ (0\leqslant x\leqslant 2)$.

Compute $\displaystyle \oint_{C} (x+y)\, dx+(x-y)\, dy$, where $C$ is the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ traversed counterclockwise.

Compute $\displaystyle \int_{C} (2a-y)\, dx+x\, dy$, where $C$ is one arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)\ (0\leqslant t\leqslant 2\pi )$.

Compute $\displaystyle \oint_{C} \dfrac {(x+y)\, dx-(x-y)\, dy}{x^{2}+y^{2}}$, where $C$ is the circle $x^{2}+y^{2}=a^{2}$ traversed counterclockwise.

Compute $\displaystyle \oint_{ABCDA} \dfrac {dx+dy}{\left|x\right|+\left|y\right|}$, where $ABCDA$ is the boundary of the square with vertices $A(1,0)$, $B(0,1)$, $C(-1,0)$, $D(0,-1)$.

Compute $\displaystyle \int_{AB} \sin y\, dx+\sin x\, dy$, where $AB$ is the straight segment between the point $A(0,\pi )$ and the point $B(\pi ,0)$.

Compute $\displaystyle \oint_{OmAnO} \arctan \dfrac {y}{x}\, dy-dx$, where $OmA$ is the parabolic arc $y=x^{2}$ and $AnO$ is the straight segment $y=x$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(-1,2)}^{(2,3)} x\, dy+y\, dx$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(0,1)}^{(3,-4)} x\, dx+y\, dy$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(0,1)}^{(2,3)} (x+y)\, dx+(x-y)\, dy$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(1,-1)}^{(1,1)} (x-y)(dx-dy)$.

Verify that the integrand is a total differential, and compute the following curve integral: $\displaystyle \int_{(0,0)}^{(a,b)} f(x+y)(dx+dy)$, where $f(u)$ is a continuous function.

Compute $\displaystyle \int_{(2,1)}^{(1,2)} \dfrac {y\, dx-x\, dy}{x^{2}}$ along a path that does not meet the $Oy$ axis.

Compute $\displaystyle \int_{(1,0)}^{(6,8)} \dfrac {x\, dx+y\, dy}{\sqrt{x^{2}+y^{2}}}$ along a path that does not pass through the origin.

Compute $\displaystyle \int_{(x_{1},y_{1})}^{(x_{2},y_{2})} \varphi (x)\, dx+\psi (y)\, dy$, where $\varphi$ and $\psi$ are continuous functions.

Compute $\displaystyle \int_{(-2,-1)}^{(3,0)} (x^{4}+4xy^{3})\, dx+(6x^{2}y^{2}-5y^{4})\, dy$.

Compute $\displaystyle \int_{(0,-1)}^{(1,0)} \dfrac {x\, dy-y\, dx}{(x-y)^{2}}$ along a path that does not meet the line $y=x$.

Compute $\displaystyle \int_{(1,\pi )}^{(2,\pi )} \left(1-\dfrac {y^{2}}{x^{2}}\cos \dfrac {y}{x}\right)dx+\left(\sin \dfrac {y}{x}+\dfrac {y}{x}\cos \dfrac {y}{x}\right)dy$ along a path that does not meet the $Oy$ axis.

Compute $\displaystyle \int_{(0,0)}^{(a,b)} e^{x}(\cos y\, dx-\sin y\, dy)$.

Prove: if $f(u)$ is a continuous function and $C$ is a piecewise smooth closed contour, then

Find the antiderivative function $z$, given $dz=(x^{2}+2xy-y^{2})\, dx+(x^{2}-2xy-y^{2})\, dy$.

Find the antiderivative function $z$, given $dz=\dfrac {y\, dx-x\, dy}{3x^{2}-2xy+3y^{2}}$.

Find the antiderivative function $z$, given $dz=\dfrac {(x^{2}+2xy+5y^{2})\, dx+(x^{2}-2xy+y^{2})\, dy}{(x+y)^{3}}$.

Find the antiderivative function $z$, given $dz=e^{x}\left[e^{y}(x-y+2)+y\right]\, dx+e^{x}\left[e^{y}(x-y)+1\right]\, dy$.

Find the antiderivative function $z$, given $dz=\dfrac {\partial^{n+m+1}u}{\partial x^{n+1}\partial y^{m}}\, dx+\dfrac {\partial^{n+m+1}u}{\partial x^{n}\partial y^{m+1}}\, dy$.

Find the antiderivative function $z$, given $dz=\dfrac {\partial^{n+m+1}}{\partial x^{n+2}\partial y^{m-1}}\left(\ln \dfrac {1}{r}\right)dx-\dfrac {\partial^{n+m+1}}{\partial x^{n-1}\partial y^{m+2}}\left(\ln \dfrac {1}{r}\right)dy$, where $r=\sqrt{x^{2}+y^{2}}$.

Prove that the following estimate holds for the curve integral:

where $L$ is the length of the path of integration, and $M=\max \sqrt{P^{2}+Q^{2}}$ (on the arc $C$).

Estimate the integral $I_{R}=\displaystyle \oint_{x^{2}+y^{2}=R^{2}} \dfrac {y\, dx-x\, dy}{(x^{2}+xy+y^{2})^{2}}$, and prove that $\displaystyle \lim_{R\to \infty } I_{R}=0$.

Compute the line integral taken along a spatial curve (assume the coordinate system is right-handed): $\displaystyle \int_{C} (y^{2}-z^{2})\, dx+2yz\, dy-x^{2}\, dz$, where $C$ is the curve $x=t$, $y=t^{2}$, $z=t^{3}\ (0\leqslant t\leqslant 1)$, traversed in the direction of increasing parameter.

Compute the line integral taken along a spatial curve (assume the coordinate system is right-handed): $\displaystyle \int_{C} y\, dx+z\, dy+x\, dz$, where $C$ is the circular helix $x=a\cos t$, $y=a\sin t$, $z=bt\ (0\leqslant t\leqslant 2\pi )$, traversed in the direction of increasing parameter.

Compute the line integral taken along the given space curve (assume the coordinate system is right-handed): $\displaystyle \int_{C} (y-z)\, dx+(z-x)\, dy+(x-y)\, dz$, where $C$ is the circle $x^{2}+y^{2}+z^{2}=a^{2}$, $y=x\tan \alpha \ (0<\alpha <\pi )$, traversed counterclockwise when viewed from the positive direction of the $Ox$ axis.

Compute the line integral taken along the given space curve (assume the coordinate system is right-handed): $\displaystyle \int_{C} y^{2}\, dx+z^{2}\, dy+x^{2}\, dz$, where $C$ is Viviani's curve $x^{2}+y^{2}+z^{2}=a^{2}$, $x^{2}+y^{2}=ax\ (z\geqslant 0,\ a>0)$, traversed counterclockwise when viewed from the positive part of the $Ox$ axis ($x>a$).

Compute the line integral taken along the given space curve (assume the coordinate system is right-handed): $\displaystyle \int_{C} (y^{2}-z^{2})\, dx+(z^{2}-x^{2})\, dy+(x^{2}-y^{2})\, dz$, where $C$ is the boundary of the part of the sphere $x^{2}+y^{2}+z^{2}=1$, $x\geqslant 0$, $y\geqslant 0$, $z\geqslant 0$, traversed in the positive direction so that the outer side of the surface remains on the left.

Using the total differential, compute the following curve integral: $\displaystyle \int_{(1,1,1)}^{(2,3,-4)} x\, dx+y^{2}\, dy-z^{3}\, dz$.

Using the total differential, compute the following curve integral: $\displaystyle \int_{(1,2,3)}^{(6,1,1)} yz\, dx+xz\, dy+xy\, dz$.

Using the total differential, compute the following curve integral: $\displaystyle \int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})} \dfrac {x\, dx+y\, dy+z\, dz}{\sqrt{x^{2}+y^{2}+z^{2}}}$, where the point $(x_{1},y_{1},z_{1})$ lies on the sphere $x^{2}+y^{2}+z^{2}=a^{2}$, and the point $(x_{2},y_{2},z_{2})$ lies on the sphere $x^{2}+y^{2}+z^{2}=b^{2}$ $(a>0,\ b>0)$.

Using the total differential, compute the following curve integral: $\displaystyle \int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})} \varphi (x)\, dx+\psi (y)\, dy+\chi (z)\, dz$, where $\varphi$, $\psi$, $\chi$ are continuous functions.

Using the total differential, compute the following curve integral: $\displaystyle \int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})} f(x+y+z)(dx+dy+dz)$, where $f$ is a continuous function.

Using the total differential, compute the following curve integral: $\displaystyle \int_{(x_{1},y_{1},z_{1})}^{(x_{2},y_{2},z_{2})} f\left(\sqrt{x^{2}+y^{2}+z^{2}}\right)(x\, dx+y\, dy+z\, dz)$, where $f$ is a continuous function.

Find the primitive function $u$, given that $du=(x^{2}-2yz)\, dx+(y^{2}-2xz)\, dy+(z^{2}-2xy)\, dz$.

Find the primitive function $u$, given that $du=\left(1-\dfrac {1}{y}+\dfrac {y}{z}\right)dx+\left(\dfrac {x}{z}+\dfrac {x}{y^{2}}\right)dy-\dfrac {xy}{z^{2}}\, dz$.

Find the primitive function $u$, given that $du=\dfrac {(x+y-z)\, dx+(x+y-z)\, dy+(x+y+z)\, dz}{x^{2}+y^{2}+z^{2}+2xy}$.

Find the work done by gravity (the $Oz$ axis pointing vertically upward) as a point of mass $m$ moves from position $(x_{1},y_{1},z_{1})$ to position $(x_{2},y_{2},z_{2})$.

An elastic force is directed toward the origin of coordinates, its magnitude proportional to the distance of the point from the origin. Suppose the point traces out, counterclockwise, one quarter of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$; find the work done by the elastic force.

Find the work done by the gravitational force $F=\dfrac {k}{r^{2}}$ (where $r=\sqrt{x^{2}+y^{2}+z^{2}}$) acting on a unit mass as it moves from the point $M_{1}(x_{1},y_{1},z_{1})$ to the point $M_{2}(x_{2},y_{2},z_{2})$.