5. Vector Calculus

Line Integrals — Quiz

Test your understanding of line integrals with 5 practice questions.

Practice Questions

Question 1

Evaluate the line integral of the scalar function $f(x, y) = x^2 + y^2$ along the curve parameterized by $\mathbf{r}(t) = (t, t^2)$ for $t$ from 0 to 1.

Question 2

Determine the work done by the vector field $\mathbf{F}(x, y) = (2x, 3y)$ along the curve $\mathbf{r}(t) = (\cos t, \sin t)$ for $t$ from 0 to $\pi$.

Question 3

Find the circulation of the vector field $\mathbf{F}(x, y) = (y, -x)$ around the unit circle $x^2 + y^2 = 1$ oriented counterclockwise.

Question 4

Compute the line integral of the vector field $\mathbf{F}(x,y) = (3x^2, 2y)$ along the line segment from $(0,0)$ to $(1,1)$.

Question 5

Evaluate the line integral $\int_C (2xy) \; ds$ where $C$ is the line segment from $(0,0)$ to $(2,1)$.