University Calculus III — Surface Integrals and Integral Theorems Worksheet

All 11 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Parametric Surfaces

   A surface $S$ is given by $$\overrightarrow{r}(u,v)=(3\cos u,v,3\sin u),0\le u\le 2\pi ,0\le v\le 4.$$

   1. Eliminate the parameters to find a Cartesian equation satisfied by every point of $S$, and name the surface, including its axis and where it begins and ends.
   2. Describe the grid curve obtained by holding $u=\frac{\pi }{2}$ fixed, and the grid curve obtained by holding $v=1$ fixed.
2. Q2Parametric Surfaces

   Find an equation of the tangent plane to the parametric surface $$\overrightarrow{r}(u,v)=(uv,u+v^2,u^2-v)$$ at the point where $u=1$ and $v=2$.

3. Q3Surface Area

   Find the area of the part of the paraboloid $z=\frac{1}{2}(x^2+y^2)$ that lies inside the cylinder $x^2+y^2=3$.

4. Q4Surface Integrals of Scalar Functions

   A thin metal panel has the shape of the part of the plane $2x+2y+z=6$ that lies in the first octant ($x\ge 0$, $y\ge 0$, $z\ge 0$), with lengths in metres. Its density at the point $(x,y,z)$ is $\sigma (x,y,z)=z$ $\text{kg}/{\text{m}}^2$. Find the mass of the panel, ${\displaystyle {\iint }_S\sigma dS}$.

5. Q5Flux Integrals of Vector Fields

   Let $S$ be the part of the paraboloid $z=4-x^2-y^2$ that lies on or above the $xy$-plane, oriented upward. Evaluate the flux ${\displaystyle {\iint }_S\overrightarrow{F}·d\overrightarrow{S}}$ of $\overrightarrow{F}(x,y,z)=(x,y,z)$ across $S$.

6. Q6Flux Integrals of Vector Fields

   Let $S$ be the cylinder $x^2+y^2=4$, $0\le z\le 3$ (the curved wall only, no top or bottom), oriented with the normal pointing away from the $z$-axis.

   1. For the parametrization $\overrightarrow{r}(\theta ,z)=(2\cos \theta ,2\sin \theta ,z)$, compute ${\overrightarrow{r}}_{\theta }\times {\overrightarrow{r}}_z$ and decide whether it agrees with the required orientation.
   2. Evaluate the flux of $\overrightarrow{F}(x,y,z)=(xz,yz,xy)$ across $S$.
7. Q7Stokes' Theorem

   Let $C$ be the curve in which the plane $x+y+z=5$ meets the cylinder $x^2+y^2=9$, traversed counterclockwise when viewed from above, and let $\overrightarrow{F}(x,y,z)=(z,x^2,y)$. Use Stokes' theorem to evaluate ${\displaystyle {\oint }_C\overrightarrow{F}·d\overrightarrow{r}}$.

8. Q8Stokes' Theorem

   Let $S$ be the part of the paraboloid $z=9-x^2-y^2$ that lies on or above the plane $z=5$, oriented upward, and let $\overrightarrow{F}(x,y,z)=(-yz,xz,xyz)$. Use Stokes' theorem to evaluate ${\displaystyle {\iint }_S(\nabla \times \overrightarrow{F})·d\overrightarrow{S}}$ as a line integral. State the boundary curve and its direction.

9. Q9The Divergence Theorem

   Let $E$ be the box $0\le x\le 2$, $0\le y\le 1$, $0\le z\le 3$, let $S$ be its surface oriented outward, and let $\overrightarrow{F}(x,y,z)=(x^2,xy,3z)$.

   1. Evaluate ${\displaystyle {\iint }_S\overrightarrow{F}·d\overrightarrow{S}}$ using the divergence theorem.
   2. Verify your answer by computing the flux through each of the six faces.
10. Q10The Divergence Theorem

    Let $E$ be the solid cylinder $x^2+y^2\le 1$, $0\le z\le 2$, and let $S$ be its complete surface (wall, top and bottom) oriented outward. Evaluate the outward flux across $S$ of $$\overrightarrow{F}(x,y,z)=(x{y}^2+\cos z,x^{2}y+e^{xz},z^2+xy).$$

11. Q11Synthesis — drawing on several topics in this unit

    Let $E$ be the solid bounded below by the paraboloid $z=x^2+y^2$ and above by the plane $z=4$. Its boundary consists of the flat top $S_1$ (the disk $x^2+y^2\le 4$ in the plane $z=4$) and the curved bowl $S_2$ (the part of $z=x^2+y^2$ with $z\le 4$), both oriented outward from $E$. Let $\overrightarrow{F}(x,y,z)=(x,y,z^2)$.

    1. State the direction (upward or downward) of the outward normal on $S_1$ and on $S_2$, and compute the flux of $\overrightarrow{F}$ across each piece.
    2. Evaluate ${\displaystyle {\iiint }_E\nabla ·\overrightarrow{F}dV}$ in cylindrical coordinates and confirm that the divergence theorem holds.

The 7 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Calculus III bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Surface Integrals and Integral Theorems notes cover  About the Surface Integrals and Integral Theorems unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

A surface is a function of two parameters

A curve needed one parameter; a surface needs two. The parametrization r(u, v) sends each point of a region in the uv-plane to a point in space, and the whole surface is what that region becomes. Q1 runs the process backwards: it hands you r(u, v) and asks what surface it is.

Part (b) is about grid curves: freeze one parameter and let the other run. Freezing u leaves a curve along which only v moves, so ask which coordinates v controls and which are now constants. Freezing v does the opposite. Each grid curve is a familiar object — a line or a circle — and naming it, with where it sits, is the whole answer.

The tangent plane comes from a cross product

Holding v fixed and differentiating in u gives r_u, a vector tangent to one grid curve; holding u fixed gives r_v, tangent to the other. Both lie in the tangent plane, so their cross product is perpendicular to it — and a point plus a normal vector is all an equation of a plane has ever needed.

Surface area is the length of that same cross product

The cross product does a second job. Its length is the area of the small parallelogram that a tiny rectangle of the parameter region is stretched into, so the area of the surface is the double integral of ‖r_u × r_v‖ over the parameter region. For a graph z = f(x, y), using x and y themselves as the parameters, that length becomes √(1 + f_x² + f_y²).

A surface integral of a density adds up mass

Q4 is the surface version of the wire in the line-integral set: a thin panel with a density σ that varies from point to point, and the mass is the integral of σ over the surface with respect to area, dS. The panel is a flat piece of a plane, which makes it the gentlest possible surface to parametrize.

Flux: orientation first, then the integral

A flux integral measures how much of a vector field passes through a surface, and it has a sign: flow in the direction of the chosen normal counts positive, flow against it negative. That is why every flux question names an orientation, and why deciding the direction of your normal vector comes before any integration.

Stokes' theorem: a line integral becomes a flux of the curl

The curve C is the boundary of the surface S, and the two orientations must agree: walk along C in its direction with your head pointing along the normal of S, and the surface is on your left. Equivalently, curl the fingers of your right hand along C and your thumb gives the normal. Green's theorem from the previous set is the special case where S is flat and lies in the xy-plane.

The divergence theorem: flux out of a closed surface

The surface must be closed — the complete boundary of a solid — and oriented outward. Then the total flux out equals the integral of the divergence over the solid inside. One triple integral replaces several surface integrals, one per face.

The synthesis question, and why it is last

Q11 closes the course by checking the divergence theorem on a solid whose boundary has two pieces: a flat lid and a curved bowl. Part (a) asks for the outward direction on each piece before any computation, and the way to decide is to ask which side of the piece the solid lies on — outward always means away from the solid, which is not the same as upward. Each flux is then a graph flux like Q5, with the sign of the area element chosen to match. Part (b) computes the divergence integral in cylindrical coordinates, with the limits of z running from the bowl up to the lid, and the two sides of the theorem must meet.

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Getting the most out of it

Decide the orientation before you integrate

Every flux, every Stokes question and every divergence-theorem question turns on a direction: upward or downward, away from the axis or towards it, outward from the solid. Write the direction you need in words, then check your normal vector against it at one convenient point. A flux with the wrong orientation is the right number with the wrong sign, and it earns nothing.

Draw the shadow

Almost every surface integral here ends as a double integral over the region the surface stands above. Sketch that region in the xy-plane — a disk, a triangle, a rectangle — and write its limits before you touch the integrand. For a disk, reach for polar coordinates and do not forget the r.

Choose the easy side of the theorem

Stokes' theorem and the divergence theorem are equalities, so each question is really a choice: which side is easier? A messy field with a simple divergence says triple integral; a curl through a curved cap says line integral round its rim; a line integral round an awkward curve says flux through the flattest surface it bounds. Say which way you are going and why — the choice is where the understanding shows.

State the hypotheses in one line

The divergence theorem needs a closed surface oriented outward; Stokes' theorem needs the curve's direction and the surface's normal to agree by the right-hand rule. Write that line every time. It costs nothing when it holds, and it is exactly what stops you applying the divergence theorem to a cylinder wall with no top or bottom.

Common questions