5. Vector Calculus

Surface Integrals — Quiz

Test your understanding of surface integrals with 5 practice questions.

Practice Questions

Question 1

Consider the parameterized surface given by $\mathbf{r}(u,v) = (u, v, u^2 + v^2)$. What is the correct expression for the surface element $dS$?

Question 2

Evaluate the surface integral $\iint_S (x + y) \, dS$ where $S$ is the surface parameterized by $\mathbf{r}(u,v) = (u, v, u^2 - v^2)$ for $0 \leq u \leq 1$ and $0 \leq v \leq 1$. What is the value of the integral?

Question 3

Which of the following is the correct form of the surface integral of a vector field $\mathbf{F}(x,y,z)$ over a surface $S$ parameterized by $\mathbf{r}(u,v)$?

Question 4

Consider the surface $S$ defined by $z = x^2 + y^2$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$. What is the integral $\iint_S z \, dS$?

Question 5

Which of the following is the correct expression for the surface integral of a scalar function $f(x,y,z)$ over a parameterized surface $\mathbf{r}(u,v)$?