08 Line Integrals Online Quiz Questions

What is the length of the curve $\mathbf r(t)=\langle 3t,4t\rangle$ for $0\le t\le 1$?

Let $C$ be the circle $x^2+y^2=4$, traversed once, and let $f(x,y)=x^2+y^2$. What is $\int_C f\,ds$?

The field $\mathbf F(x,y)=\langle 6x,2y\rangle$ acts along any curve from $(0,0)$ to $(1,2)$. What is $\int_C\mathbf F\cdot d\mathbf r$?

Select all statements that are correct about line integrals and orientation.

Select all formulas or statements that correctly describe the evaluation of line integrals.

True or false: Reversing the orientation of a curve leaves the value of a scalar line integral $\int_C f\,ds$ unchanged.

True or false: If $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$ everywhere that a planar field is defined, then the field is automatically conservative on its entire domain.

Compute $\int_C\mathbf F\cdot d\mathbf r$, where $\mathbf F(x,y)=\langle x,2y\rangle$ and $C$ is parametrized by $\mathbf r(t)=\langle t,t^2\rangle$ for $0\le t\le 1$. Enter the numerical value.

What is the standard term for a vector field that equals the gradient of a scalar potential function and therefore has path-independent line integrals?

Let $\mathbf F(x,y)=\langle 2x,6y\rangle$, and let $C$ be any piecewise smooth path from $(1,0)$ to $(2,1)$. Use a potential function to compute $\int_C\mathbf F\cdot d\mathbf r$, and explain why the result does not depend on the chosen path.

A scalar line integral $\int_C f\,ds$ is evaluated along a curve exactly once. What happens if the direction of traversal is reversed?

True or false: Reversing the orientation of a curve changes the sign of a vector line integral $\int_C\mathbf F\cdot d\mathbf r$.

For the parametrized curve $\mathbf r(t)=\langle 3t,4t\rangle$, $0\le t\le 1$, which expression gives the differential of arc length $ds$?

Let $C$ be the circle $x^2+y^2=9$, traversed once counterclockwise, and let $f(x,y)=x^2+y^2$. What is $\int_C f\,ds$?

Compute the work done by $\mathbf F(x,y)=\langle y,x\rangle$ along the line segment from $(0,0)$ to $(2,1)$, oriented from the first point to the second.

Consider $\mathbf F(x,y)=\langle x^2+y^2,\,2xy\rangle$ on $\mathbb R^2$. Which conclusion is justified by the conservative-field test?

The vector field $\mathbf F(x,y)=\langle 2x,2y\rangle$ moves a particle from $(1,0)$ to $(3,2)$ along any smooth path. Using a potential function, determine the exact work.

What is the standard term for a vector field $\mathbf F$ that can be written as $\mathbf F=\nabla f$ for some scalar function $f$?