Free Practice Quiz Question List

10 Parametric Surfaces and Surface Integrals Online Quiz Questions

Use this free practice quiz with 20 questions to review 10 Parametric Surfaces and Surface Integrals, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: A parametrization $\mathbf r(u,v)$ is regular at a point when $\mathbf r_u\times\mathbf r_v\ne\mathbf 0$ at that point.

A
True
B
False

Choose one

1 point

Which quantity is required in the surface-area element $dS$ for a parametrized surface?

A
Use $\mathbf r_u\times\mathbf r_v$ directly.
B
Use $\|\mathbf r_u\times\mathbf r_v\|$ .
C
Use $\mathbf r_u+\mathbf r_v$ .
D
Use $\mathbf r_u\cdot\mathbf r_v$ .

Written response

1 point

For the graph $z=f(x,y)=x+2y$ , what is the value of $f_x$ ?

Fill in the blank

1 point

In a parametrized surface $\mathbf r(u,v)$ , the region $D$ is the , and $\mathbf r_u$ and $\mathbf r_v$ are the two .

Choose one

1 point

For a graph $z=f(x,y)$ , which vector area element gives the upward orientation?

A
$\langle -f_x,-f_y,1\rangle\,dx\,dy$
B
$\langle f_x,f_y,-1\rangle\,dx\,dy$
C
$\langle f_x,f_y,1\rangle\,dx\,dy$
D
$\langle -f_x,-f_y,-1\rangle\,dx\,dy$

True or false

1 point

True or false: Reversing the orientation of a surface changes the value of a scalar surface integral.

A
True
B
False

Choose all

1 point

Select all correct statements about $\mathbf r_u\times\mathbf r_v$ for a parametrized surface.

A
$\mathbf r_u\times\mathbf r_v$ is normal to the surface.
B
$\mathbf r_u\times\mathbf r_v$ must always have unit length.
C
$\|\mathbf r_u\times\mathbf r_v\|$ supplies the area-scaling factor.
D
A scalar surface integral changes sign when the orientation is reversed.

Written response

1 point

What is the standard term for the region $D$ in the $uv$ -plane over which the parameters $(u,v)$ vary?

Fill in the blank

1 point

For a parametrized surface, the scalar surface-integral area factor is , whereas the oriented vector area element used for flux is .

Choose one

1 point

For the spherical parametrization $\mathbf r(\phi,\theta)=\langle a\sin\phi\cos\theta,a\sin\phi\sin\theta,a\cos\phi\rangle$ , which parameter domain covers the sphere?

A
$0\le\phi\le\pi$ and $0\le\theta\le 2\pi$
B
$0\le\phi\le 2\pi$ and $0\le\theta\le\pi$
C
$-\pi\le\phi\le\pi$ and $0\le\theta\le 2\pi$
D
$0\le\phi\le\pi/2$ and $0\le\theta\le\pi$

Choose all

1 point

Select all correct statements about evaluating a surface or flux integral.

A
Compute $\mathbf r_u$ and $\mathbf r_v$ .
B
Use $\|\mathbf r_u\times\mathbf r_v\|$ for every surface integral, including flux.
C
Check whether the cross-product direction matches the required orientation.
D
Substitute the parametrization into the integrand before integrating.

Open ended

1 point

Derive the upward flux of $\mathbf F=\langle x,y,z\rangle$ through the portion of the plane $z=1-x-y$ above $D=\{(x,y):x\ge0,\ y\ge0,\ x+y\le1\}$ , and evaluate the flux.

Choose one

1 point

Which parametrization represents the portion of the graph $z=x^2+y$ over a region $D$ in the $xy$ -plane?

A
$\mathbf r(x,y)=\langle x,y,x^2+y\rangle$
B
$\mathbf r(x,y)=\langle x^2+y,x,y\rangle$
C
$\mathbf r(x,y)=\langle x,y,x^2y\rangle$
D
$\mathbf r(x,y)=\langle x^2,y,x+y\rangle$

Choose one

1 point

For the plane $z=2x-y$ , what factor multiplies $dA$ in the surface-area integral over a region $D$ in the $xy$ -plane?

A
$1$
B
$\sqrt 2$
C
$\sqrt 6$
D
$2$

Written response

1 point

For a parametrized surface $\mathbf r(u,v)$ , what is the name of the region $D$ in the $uv$ -plane over which $(u,v)$ varies?

Choose one

1 point

The surface is the graph $z=x^2+y$ . Which vector should be used as the upward-oriented vector area element in a flux integral?

A
$\langle 2x,1,1\rangle$
B
$\langle -2x,-1,1\rangle$
C
$\langle -2x,1,-1\rangle$
D
$\langle 2x,-1,-1\rangle$

Written response

1 point

Find the upward flux of $\mathbf F=\langle 0,0,z\rangle$ through the graph $z=x+y$ over the unit square $0\le x\le 1$ , $0\le y\le 1$ . Enter the exact numerical value.

Choose one

1 point

Which parametrization and parameter bounds describe the lateral surface of the cylinder of radius $3$ between $z=0$ and $z=5$ ?

A
$\mathbf r(\theta,z)=\langle 3\cos\theta,3\sin\theta,z\rangle$ , $0\le\theta\le2\pi$ , $0\le z\le5$
B
$\mathbf r(\theta,z)=\langle 5\cos\theta,5\sin\theta,z\rangle$ , $0\le\theta\le2\pi$ , $0\le z\le3$
C
$\mathbf r(\theta,z)=\langle 3\sin\theta,3 heta,z\rangle$ , $0\le\theta\le5$ , $0\le z\le2\pi$
D
$\mathbf r(\theta,z)=\langle 3\cos z,3\sin z,\theta\rangle$ , $0\le\theta\le5$ , $0\le z\le2\pi$

Choose one

1 point

Is the parametrization $\mathbf r(u,v)=\langle u^2,v,u+v\rangle$ regular at $(u,v)=(0,0)$ ?

A
Yes, because $\mathbf r_u\times\mathbf r_v\ne\mathbf 0$ at $(0,0)$
B
No, because one component of $\mathbf r_u$ is zero at $(0,0)$
C
No, because the parametrization contains the term $u^2$
D
Yes, because every parametrization is regular everywhere

True or false

1 point

True or false: The mass of the surface $z=x+y$ over the unit square $0\le x\le 1$ , $0\le y\le 1$ , when the surface density is $\rho(x,y,z)=z$ , is $\sqrt{3}$ .

A
True
B
False