Lesson 31: Surface integrals - I $17.6$

Accessible transcription generated on 4/6/2026

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Lesson 31: Surface integrals - I $17.6$

Warmup: What is a surface?

Conceptual diagram titled 'stack lines' showing five parallel blue line segments slanted diagonally upwards from left to right. It illustrates the idea that a surface can be visualized as a continuous stack of individual lines.

Visual Description: Conceptual diagram titled 'stack lines' showing five parallel blue line segments slanted diagonally upwards from left to right. It illustrates the idea that a surface can be visualized as a continuous stack of individual lines.

Conceptual diagram titled 'stack curves' showing three parallel blue wavy curves stacked vertically. This represents the concept of a surface being formed by a continuous sequence of curves.

Visual Description: Conceptual diagram titled 'stack curves' showing three parallel blue wavy curves stacked vertically. This represents the concept of a surface being formed by a continuous sequence of curves.

stack lines stack curves

eg: plane, sphere, cone, paraboloid

A collection of four hand-drawn sketches illustrating common mathematical surfaces. Top left: A plane represented as a tilted blue parallelogram with parallel hatching lines. Bottom left: A sphere shown as a circle with horizontal elliptical cross-sections (solid lines in front, dashed in back). Bottom center: An inverted cone with its vertex at the bottom and horizontal circular slices drawn inside. Bottom right: An upward-opening paraboloid (bowl shape) with horizontal circular cross-sections used to indicate its three-dimensional volume.

Visual Description: A collection of four hand-drawn sketches illustrating common mathematical surfaces. Top left: A plane represented as a tilted blue parallelogram with parallel hatching lines. Bottom left: A sphere shown as a circle with horizontal elliptical cross-sections (solid lines in front, dashed in back). Bottom center: An inverted cone with its vertex at the bottom and horizontal circular slices drawn inside. Bottom right: An upward-opening paraboloid (bowl shape) with horizontal circular cross-sections used to indicate its three-dimensional volume.

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Curve

Definition: A curve is a 1D object.

Representations of a curve:

Graph of $y = f(x)$
Example: $y = x^2$

The diagram displays a standard Cartesian coordinate system with a smooth, symmetrical parabola opening upwards, vertex at the origin (0,0), representing the quadratic function $y = x^2$.
Level curve of $f(x,y)$
Function: $z = f(x,y) = x^2 - y$

Level set: $z = 0 \implies y = x^2$, which is a parabola.
Parametric Curves
Representation: $\vec{r}(t)$

Surface

Definition: A surface is a 2D object.

Representations of a surface:

Graph of $z = f(x,y)$
Example: $z = x^2 + y^2$

The diagram shows a three-dimensional perspective plot of a circular paraboloid. The surface is centered at the origin (0,0,0) and opens upward along the positive z-axis, resembling a bowl.
Level surface of $f(x,y,z)$
Function: $w = f(x,y,z) = x^2 + y^2 - z$

Level set: $w = 0 \implies z = x^2 + y^2$, which is a paraboloid.
Parametric Surfaces
Representation: $\vec{r}(u,v)$

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Parametric Curves

$A conceptual diagram illustrating the mapping of a parametric curve from a 1D domain to 3D space. At the top, a horizontal line represents the parameter 't' axis, with a segment bounded by points 'a' and 'b'. Below this is a 3D Cartesian coordinate system with three axes. Three color-coded arrows (purple, green, and red) demonstrate the mapping from the 1D parameter space to 3D space: the purple arrow maps point 'a' to the starting point of a blue curve; the red arrow maps point 'b' to the ending point of the curve; and the green arrow maps an intermediate point 't' to a specific point on the curve labeled \vec{r}(t). This visualizes how a single parameter defines a path in space.$

Visual Description: A conceptual diagram illustrating the mapping of a parametric curve from a 1D domain to 3D space. At the top, a horizontal line represents the parameter 't' axis, with a segment bounded by points 'a' and 'b'. Below this is a 3D Cartesian coordinate system with three axes. Three color-coded arrows (purple, green, and red) demonstrate the mapping from the 1D parameter space to 3D space: the purple arrow maps point 'a' to the starting point of a blue curve; the red arrow maps point 'b' to the ending point of the curve; and the green arrow maps an intermediate point 't' to a specific point on the curve labeled \vec{r}(t). This visualizes how a single parameter defines a path in space.

\[\vec{r}(t) = \langle x(t), y(t), z(t) \rangle\] \[a \leq t \leq b\]

\[\vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle\]

Parametric Surfaces

A conceptual diagram illustrating the mapping of a parametric surface from a 2D domain to 3D space. At the top, a 2D Cartesian plane with axes labeled 'u' and 'v' contains an irregular shaded region labeled 'R'. Below this is a 3D Cartesian coordinate system. Three color-coded arrows (purple, green, and red) map representative points from within the 2D region 'R' down to corresponding points on a blue shaded surface labeled 'S' in the 3D space. This visualizes the transformation of a 2D parameter area into a curved 3D surface where each point on the surface is determined by a unique pair of parameters (u, v).

Visual Description: A conceptual diagram illustrating the mapping of a parametric surface from a 2D domain to 3D space. At the top, a 2D Cartesian plane with axes labeled 'u' and 'v' contains an irregular shaded region labeled 'R'. Below this is a 3D Cartesian coordinate system. Three color-coded arrows (purple, green, and red) map representative points from within the 2D region 'R' down to corresponding points on a blue shaded surface labeled 'S' in the 3D space. This visualizes the transformation of a 2D parameter area into a curved 3D surface where each point on the surface is determined by a unique pair of parameters (u, v).

\[\vec{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle\]

$u, v$ lie in $R$ in $\mathbb{R}^2$

bounds for $u, v$

\[\vec{r}_u = \langle x_u, y_u, z_u \rangle\] \[\vec{r}_v = \langle x_v, y_v, z_v \rangle\]

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Example: Parameterized Surface

\[\vec{r}(u,v) = \langle \cos u, \sin u, 2v \rangle, \quad 0 \le u \le \frac{3\pi}{2}, \quad 0 \le v \le 1.\]

A graph in the uv-coordinate plane illustrating the parameter domain. The horizontal axis is labeled 'u' and the vertical axis is labeled 'v'. The domain is a rectangle bounded by u from 0 to 3π/2 and v from 0 to 1. The horizontal line at v=0 is highlighted in purple, the line at v=1 is highlighted in red, and an intermediate horizontal line at a generic value of v is highlighted in green.

Visual Description: A graph in the uv-coordinate plane illustrating the parameter domain. The horizontal axis is labeled 'u' and the vertical axis is labeled 'v'. The domain is a rectangle bounded by u from 0 to 3π/2 and v from 0 to 1. The horizontal line at v=0 is highlighted in purple, the line at v=1 is highlighted in red, and an intermediate horizontal line at a generic value of v is highlighted in green.

$v=0, \ 0 \le u \le \frac{3\pi}{2}$
$\vec{r}(u,0) = \langle \cos u, \sin u, 0 \rangle$

$v=1, \ 0 \le u \le \frac{3\pi}{2}$
$\vec{r}(u,1) = \langle \cos u, \sin u, 2 \rangle$

fix $v: 0 \le v \le 1$
$\vec{r}(u, "v") = \langle \cos u, \sin u, "2v" \rangle$

A 3D coordinate plot in xyz-space showing the surface defined by the parameterization. The resulting shape is a three-quarter cylinder with a radius of 1. It is shown as a vertical stack of arcs: a purple base arc in the xy-plane (z=0), a red top arc at height z=2, and a green arc at an intermediate height. The arcs span the angular range from 0 to 3π/2. Dashed lines represent the negative parts of the coordinate axes.

Visual Description: A 3D coordinate plot in xyz-space showing the surface defined by the parameterization. The resulting shape is a three-quarter cylinder with a radius of 1. It is shown as a vertical stack of arcs: a purple base arc in the xy-plane (z=0), a red top arc at height z=2, and a green arc at an intermediate height. The arcs span the angular range from 0 to 3π/2. Dashed lines represent the negative parts of the coordinate axes.

stack $\frac{3}{4}$th circles of radius 1 from $z=0$ to $z=2$
\[\downarrow\]
$\frac{3}{4}$th cylinder of radius 1 height 2

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Parametrizing a Graph $z = f(x, y)$

Recall: $y = x^2$ from $(0, 0)$ to $(2, 4)$

A hand-drawn graph showing a 2D Cartesian coordinate system with vertical y and horizontal x axes. A parabolic curve representing y = x^2 starts at the origin (0,0) and extends into the first quadrant. An arrow on the curve indicates the direction of motion as the parameter t increases from 0 toward 2, corresponding to the point (2, 4).

Visual Description: A hand-drawn graph showing a 2D Cartesian coordinate system with vertical y and horizontal x axes. A parabolic curve representing y = x^2 starts at the origin (0,0) and extends into the first quadrant. An arrow on the curve indicates the direction of motion as the parameter t increases from 0 toward 2, corresponding to the point (2, 4).

parametrization $\vec{r}(t) = \langle t, t^2 \rangle, 0 \le t \le 2$ \[x = t, \quad y = x^2 = t^2\]

To parametrize a surface defined by the graph $z = f(x, y)$: \[z = f(x, y) \leadsto x = u, \quad y = v, \quad z = f(u, v)\] The vector-valued function is given by: \[\vec{r}(u, v) = \langle u, v, f(u, v) \rangle\]

Additionally, define Bounds for $u, v$ based on the Domain or a part of the domain of $f$.

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Example: Parametrizing a Paraboloid

$A 3D coordinate plot showing a paraboloid opening upwards from the origin. The surface is defined by the equation $z = x^2 + y^2$. Horizontal dashed ellipses are drawn on the surface at various heights to indicate circular cross-sections, illustrating the shape of the surface in 3D space.$

Visual Description: A 3D coordinate plot showing a paraboloid opening upwards from the origin. The surface is defined by the equation $z = x^2 + y^2$. Horizontal dashed ellipses are drawn on the surface at various heights to indicate circular cross-sections, illustrating the shape of the surface in 3D space.

Consider the surface defined by the equation:

\[z = x^2 + y^2\]

Parametrization:

We can parameterize this surface using $u$ and $v$ as follows:

\[x = u, \quad y = v, \quad z = u^2 + v^2\]

This gives the vector-valued function:

\[\vec{r}(u,v) = \langle u, v, u^2 + v^2 \rangle\]

Bounds:

\[-\infty < u < \infty\] \[-\infty < v < \infty\]

What if we are only interested in the part of the surface between $1 \le z \le 9$?

Substituting our parametrization for $z$, we get the condition:

\[1 \le z = u^2 + v^2 \le 9\]

$A mapping diagram illustrating the relationship between the parameter domain and the resulting surface section. On the left, a 2D graph in the $uv$-plane shows an annular region (a ring) centered at the origin, shaded with blue diagonal lines and labeled $R$. The boundaries of this region correspond to $u^2 + v^2 = 1$ and $u^2 + v^2 = 9$. On the right, a 3D graph shows the section of the paraboloid $z = x^2 + y^2$ that lies between $z=1$ and $z=9$, also shaded with blue diagonal lines and labeled $S$. A red curved arrow points from the region $R$ in the $uv$-plane to the surface $S$ in 3D space, demonstrating how the 2D region is mapped onto the 3D surface.$

Visual Description: A mapping diagram illustrating the relationship between the parameter domain and the resulting surface section. On the left, a 2D graph in the $uv$-plane shows an annular region (a ring) centered at the origin, shaded with blue diagonal lines and labeled $R$. The boundaries of this region correspond to $u^2 + v^2 = 1$ and $u^2 + v^2 = 9$. On the right, a 3D graph shows the section of the paraboloid $z = x^2 + y^2$ that lies between $z=1$ and $z=9$, also shaded with blue diagonal lines and labeled $S$. A red curved arrow points from the region $R$ in the $uv$-plane to the surface $S$ in 3D space, demonstrating how the 2D region is mapped onto the 3D surface.

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Example: Parameterizing a Plane

Consider the equation of a plane: \[4x + 3y + z = 12\] We can solve for $z$ to get: \[z = 12 - 4x - 3y\] The parameterization is then defined by: \[\vec{r}(u,v) = \langle u, v, 12 - 4u - 3v \rangle\]

For the entire plane, the bounds are: \[-\infty < u < \infty\] \[-\infty < v < \infty\]

Restricting the Surface

What if we want only the part of the plane in the 1st Octant?

In the first octant, the coordinates must satisfy: \[x \ge 0, \quad y \ge 0, \quad z \ge 0\] Translating these conditions into our parameters $u$ and $v$, we get: \[u \ge 0, \quad v \ge 0, \quad 12 - 4u - 3v \ge 0\]

A two-part diagram illustrating the mapping from parameter space to 3D space. On the left, there is a 2D coordinate system with axes u and v. A line segment is drawn between (3,0) on the u-axis and (0,4) on the v-axis, labeled with the equation 4u + 3v = 12. The triangular region bounded by the u-axis, v-axis, and this line is shaded with blue diagonal lines and labeled 'R'. On the right, there is a 3D coordinate system showing a triangular surface 'S' in the first octant. The vertices of this triangle are at (3,0,0) on the x-axis, (0,4,0) on the y-axis, and (0,0,12) on the z-axis. A large red arrow points from the shaded region R in the 2D plane to the surface S in the 3D space, representing the vector-valued function r(u,v).

Visual Description: A two-part diagram illustrating the mapping from parameter space to 3D space. On the left, there is a 2D coordinate system with axes u and v. A line segment is drawn between (3,0) on the u-axis and (0,4) on the v-axis, labeled with the equation 4u + 3v = 12. The triangular region bounded by the u-axis, v-axis, and this line is shaded with blue diagonal lines and labeled 'R'. On the right, there is a 3D coordinate system showing a triangular surface 'S' in the first octant. The vertices of this triangle are at (3,0,0) on the x-axis, (0,4,0) on the y-axis, and (0,0,12) on the z-axis. A large red arrow points from the shaded region R in the 2D plane to the surface S in the 3D space, representing the vector-valued function r(u,v).

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Parametrization of a Sphere

Example: Parametrize a sphere of radius 2, centered at the origin.

The Cartesian equation for a sphere of radius 2 is:

\[x^2 + y^2 + z^2 = 4\]

Solving for $z$, we can describe the surface in two parts:

Top: $z = \sqrt{4 - x^2 - y^2}$
Bottom: $z = -\sqrt{4 - x^2 - y^2}$

Alternatively, this surface is much simpler in spherical coordinates, where the radius $\rho$ is constant:

\[\rho = 2\]

A 3D Cartesian coordinate system diagram illustrating the relationship between Cartesian (x, y, z) and spherical coordinates (rho, phi, theta). A vector of length rho extends from the origin to a point labeled (x, y, z). The angle phi is shown as the angle between the positive z-axis and the vector. The angle theta is shown in the xy-plane, measured from the positive x-axis to the projection of the vector onto that plane. Dashed lines illustrate the projection of the point onto the z-axis and the xy-plane to show the geometric components used in the conversion formulas.

Visual Description: A 3D Cartesian coordinate system diagram illustrating the relationship between Cartesian (x, y, z) and spherical coordinates (rho, phi, theta). A vector of length rho extends from the origin to a point labeled (x, y, z). The angle phi is shown as the angle between the positive z-axis and the vector. The angle theta is shown in the xy-plane, measured from the positive x-axis to the projection of the vector onto that plane. Dashed lines illustrate the projection of the point onto the z-axis and the xy-plane to show the geometric components used in the conversion formulas.

The standard equations for converting from spherical to Cartesian coordinates are:

\[x = \rho \sin \phi \cos \theta\] \[y = \rho \sin \phi \sin \theta\] \[z = \rho \cos \phi\]

By substituting $\rho = 2$, we obtain the vector parametrization of the sphere:

\[\mathbf{r}(\phi, \theta) = \langle 2 \sin \phi \cos \theta, 2 \sin \phi \sin \theta, 2 \cos \phi \rangle\]

The bounds for the parameters to cover the entire sphere are:

\[0 \le \theta \le 2\pi\] \[0 \le \phi \le \pi\]

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Surface integral of a function $f(x,y,z)$

The following notes describe the formulation of surface integrals, using the analogy of line integrals for context.

Review: Parameterization of Line Integrals

$Diagram illustrating the parameterization of a curve for a line integral. At the top, a horizontal line represents the parameter space with an interval marked by points 'a' and 'b'. A small red segment labeled 'dt' represents an infinitesimal change in the parameter 't'. A downward-pointing arrow labeled '$\vec{r}(t)$' shows the mapping from this interval to a blue curve 'C' in space. On the curve 'C', a corresponding red segment 'ds' represents the infinitesimal arc length.$

Visual Description: Diagram illustrating the parameterization of a curve for a line integral. At the top, a horizontal line represents the parameter space with an interval marked by points 'a' and 'b'. A small red segment labeled 'dt' represents an infinitesimal change in the parameter 't'. A downward-pointing arrow labeled '$\vec{r}(t)$' shows the mapping from this interval to a blue curve 'C' in space. On the curve 'C', a corresponding red segment 'ds' represents the infinitesimal arc length.

For a line integral over a curve $C$:

\[ \int_C f \, ds \]

The infinitesimal arc length $ds$ is defined as:

\[ ds = |\vec{r}'(t)| \, dt \]

The integral over the curve is evaluated over the interval $[a, b]$ as:

\[ \int_C f \, ds = \int_a^b f(\vec{r}(t)) |\vec{r}'(t)| \, dt \]

In the case where the function $f = 1$, the integral yields the total length of the curve:

\[ \int_C 1 \, ds = \text{Arc length} \]

Definition: Surface Integral

$Diagram illustrating the parameterization of a surface. On the left, a region 'R' is shaded in blue within a 2D coordinate system representing the $uv$-plane. A small red element inside 'R' is labeled 'dA'. An arrow labeled '$\vec{r}(u,v)$' points from this region to a blue curved surface 'S' in 3D space on the right. A corresponding small red element on surface 'S' is labeled 'dS', which is identified in text as the infinitesimal surface area.$

Visual Description: Diagram illustrating the parameterization of a surface. On the left, a region 'R' is shaded in blue within a 2D coordinate system representing the $uv$-plane. A small red element inside 'R' is labeled 'dA'. An arrow labeled '$\vec{r}(u,v)$' points from this region to a blue curved surface 'S' in 3D space on the right. A corresponding small red element on surface 'S' is labeled 'dS', which is identified in text as the infinitesimal surface area.

The surface integral of a function $f$ over a surface $S$ is expressed as:

\[ \iint_S f \, dS \]

In this expression, $dS$ represents the infinitesimal area element, which is calculated using the magnitude of the cross product of the partial derivatives of the parameterization $\vec{r}(u,v)$:

\[ dS = |\vec{r}_u \times \vec{r}_v| \, dA \]

General Formula: The formula for evaluating the surface integral over the parameter region $R$ is:

\[ \iint_S f \, dS = \iint_R f(\vec{r}(u,v)) |\vec{r}_u \times \vec{r}_v| \, dA \]

When the function $f = 1$, the integral represents the total area of the surface $S$:

\[ \iint_S 1 \, dS = \text{Surface Area} \]

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Surface Area Example

Find the area of the part of the plane $4x + 3y + z = 12$ in the $1^{\text{st}}$ octant.

Solution:

\[\vec{r}(u, v) = \langle u, v, 12 - 4u - 3v \rangle\]

where $u, v$ are in $R$.

A technical diagram illustrating the parametrization mapping of a surface. On the left, there is a 2D graph of the region R in the uv-plane. It shows a right-angled triangle in the first quadrant shaded with blue diagonal lines. The vertices of the triangle are at the origin (0,0), on the u-axis at (3,0), and on the v-axis at (0,4). The hypotenuse is labeled with the equation 4u + 3v = 12. On the right, there is a 3D coordinate system showing the triangular surface S in the first octant of xyz-space. A red curved arrow labeled r(u,v) points from the region R in the uv-plane to the surface S in the 3D space, indicating the transformation between the parameter space and the physical surface.

Visual Description: A technical diagram illustrating the parametrization mapping of a surface. On the left, there is a 2D graph of the region R in the uv-plane. It shows a right-angled triangle in the first quadrant shaded with blue diagonal lines. The vertices of the triangle are at the origin (0,0), on the u-axis at (3,0), and on the v-axis at (0,4). The hypotenuse is labeled with the equation 4u + 3v = 12. On the right, there is a 3D coordinate system showing the triangular surface S in the first octant of xyz-space. A red curved arrow labeled r(u,v) points from the region R in the uv-plane to the surface S in the 3D space, indicating the transformation between the parameter space and the physical surface.

\[\text{Area} = \iint_S 1 \, dS = \iint_R |\vec{r}_u \times \vec{r}_v| \, dA\]

Calculating the partial derivatives and their cross product:

\[\vec{r}_u = \langle 1, 0, -4 \rangle\] \[\vec{r}_v = \langle 0, 1, -3 \rangle\] \[\vec{r}_u \times \vec{r}_v = \langle 4, 3, 1 \rangle, \quad |\vec{r}_u \times \vec{r}_v| = \sqrt{26}\]

The integral becomes:

\[\text{Area} = \iint_R \sqrt{26} \, dA = \sqrt{26} \iint_R 1 \, dA\] \[\text{Area} = \sqrt{26} \times \text{Area of } R = 6\sqrt{26}.\]

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eg: find area of surface parametrized by \[ \vec{r}(u,v) = \langle \cos u, \sin u, 2v \rangle, \quad 0 \le u \le \frac{3\pi}{2}, \quad 0 \le v \le 1 \]

Lesson 31: Surface integrals - I \(17.6\)