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 $t$ in $[0, 1]$.

Question 2

Consider the vector field $\mathbf{F}(x,y) = (2x, 3y)$. Compute the circulation (line integral) of $\mathbf{F}$ around the closed curve $C$, where $C$ is the circle $x^2 + y^2 = 4$ oriented counterclockwise.

Question 3

Find the value of the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F}(x,y) = (-y, x)$ and $C$ is the line segment from $(1,0)$ to $(0,1)$.

Question 4

Evaluate the line integral of the vector field $\mathbf{F}(x,y) = (y, -x)$ along the curve given by $\mathbf{r}(t) = (\cos t, \sin t)$ for $t \in [0, \pi]$.

Question 5

Compute the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F}(x,y) = (y^2, x^2)$ and $C$ is the curve parameterized by $\mathbf{r}(t) = (t, t^2)$ for $t \in [0, 1]$.