Multiple Integrals - Theory & Concepts

Multiple Integrals

$Learning Objectives$

Interpret a double integral as signed accumulation over a planar region.
Evaluate double integrals by iterated integration and Fubini's Theorem.
Describe Type I and Type II regions and reverse the order of integration when advantageous.
Convert circular and sector-shaped regions to polar coordinates using $dA=r\,dr\,d\theta$ .
Apply double integrals to area, volume under a surface, mass, first moments, centroids, and planar moments of inertia.
Set up introductory surface-area integrals for smooth graphs and parametrized surfaces.
Distinguish an ordinary region integral from an iterated integral written with reversed (oriented) limits.

A double integral accumulates a scalar quantity over a two-dimensional region. Depending on the integrand, that accumulation can represent signed volume, area, mass, a first moment, or another engineering quantity. This topic develops the planar geometry and coordinate skills needed before triple integrals and general changes of variables are introduced later.

Double Integrals and Signed Accumulation

$Double Integral$

For an integrable function $f$ on a planar region $R$ , the double integral $\iint_R f(x,y)\,dA$ is the limit of signed Riemann sums $\sum f(x_i^*,y_j^*)\,\Delta A$ .

When $f(x,y)\ge 0$ , $\iint_R f\,dA$ can be interpreted as the geometric volume between $z=f(x,y)$ and the $xy$ -plane. When $f$ changes sign, contributions above the plane are positive and contributions below the plane are negative. The integral is therefore a signed quantity, not the sum of absolute box volumes.

Double Integral

Signed accumulation of f over a planar region R.

\iint_R f(x,y)\,dA

Variables

Symbol	Description	Unit
$R$	Planar region of integration	-
$f(x,y)$	Scalar integrand	-
$dA$	Differential area element	-

$Interactive Simulation$

The simulation below uses midpoint Riemann sums on the square $[-2,2]\times[-2,2]$ . Boxes below the $xy$ -plane contribute negatively, so the displayed approximation represents the signed double integral over the entire square.

Signed Double Integral Visualizer

Function z = f(x, y)

Grid resolution: 10 × 10

Midpoint cells above the xy-plane contribute positively; cells below it contribute negatively. Increasing the resolution improves the Riemann-sum approximation.

Signed integral over [-2, 2] × [-2, 2]

21.7600

Exact value: 21.3333
Absolute error: 4.27e-1

\iint_R 4-x^2-y^2\,dA \approx \sum f(x_i^*,y_j^*)\,\Delta A

Positive contributionNegative contribution

Drag to rotate • Scroll to zoom

$Geometric volume versus signed integral$

If the goal is the geometric volume between a surface and the plane and the surface crosses the plane, split the region where $f$ changes sign and integrate $|f|$ , or otherwise account for the positive and negative portions separately. By contrast, $\iint_R f\,dA$ itself keeps the sign of $f$ .

Iterated Integrals and Fubini's Theorem

$Iterated Integral$

An iterated integral evaluates a multiple integral through successive single-variable integrations, treating variables not currently integrated as constants.

$Fubini's Theorem on a rectangle$

If $f$ is continuous on $R=[a,b]\times[c,d]$ , then

\iint_R f(x,y)\,dA = \int_a^b\int_c^d f(x,y)\,dy\,dx = \int_c^d\int_a^b f(x,y)\,dx\,dy.

The same conclusion holds under broader integrability hypotheses; continuity is a convenient sufficient condition for this course.

For a rectangle, reversing the order only swaps which constant interval is integrated first. For a nonrectangular region, reversing the order requires reconstructing the region's boundary equations.

Type I and Type II Regions

$Type I region: vertical slices$

A Type I region can be written

R=\{(x,y):a\le x\le b,\ g_1(x)\le y\le g_2(x)\},

with $g_1(x)\le g_2(x)$ throughout $[a,b]$ .

Type I Integral

Integrate with respect to y first.

\iint_R f(x,y)\,dA = \int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx.

$Type II region: horizontal slices$

A Type II region can be written

R=\{(x,y):c\le y\le d,\ h_1(y)\le x\le h_2(y)\},

with $h_1(y)\le h_2(y)$ throughout $[c,d]$ .

Type II Integral

Integrate with respect to x first.

\iint_R f(x,y)\,dA = \int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy.

$Changing the Order of Integration$

STEP-BY-STEP

Sketch or analyze the region described by the original bounds.
Identify its projection onto the new outer-variable axis.
For each value of the new outer variable, determine the left/right or lower/upper boundary for the new inner integral.
Split the region if a single pair of boundary functions cannot describe every slice.
Check that the rebuilt bounds cover exactly the same planar set.

$Reversed inner limits are oriented limits, not an ordinary Type I description$

An expression such as

\int_0^1\int_x^{x^2} f(x,y)\,dy\,dx

is algebraically valid. However, for $0<x<1$ we have $x^2<x$ , so the inner limits are reversed. It equals

-\int_0^1\int_{x^2}^{x} f(x,y)\,dy\,dx.

The second form is the ordinary Type I region integral; the first is its negative because of orientation.

Polar Double Integrals

Polar coordinates are often advantageous when circles, annuli, radial functions, or angular sectors dominate the geometry. Use $x=r\cos\theta$ , $y=r\sin\theta$ , and $r\ge 0$ .

$Polar area scaling$

A small polar cell has radial thickness $dr$ and approximate arc length $r\,d\theta$ . Its differential area is therefore

dA=r\,dr\,d\theta.

The factor $r$ is the area-scaling factor of the polar coordinate transformation. A later topic develops the full Jacobian change-of-variables theorem.

Polar Double Integral

Double integral after converting the planar region to polar coordinates.

\iint_R f(x,y)\,dA = \int_{\alpha}^{\beta} \int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta.

$Polar Setup$

STEP-BY-STEP

Rewrite the region in terms of $r$ and $\theta$ .
Rewrite every occurrence of $x$ and $y$ in the integrand.
Include the factor $r$ in $dA$ .
Confirm the angular interval sweeps the intended region exactly once.
Check dimensions: $r\,dr\,d\theta$ has units of area.

$Interactive Simulation$

The polar simulation shows a sector and a differential annular wedge. Keep the wedge inside the active region while observing how its area grows with radius.

Integrand function

Radius limit (

R_{max}

): 3.0 m

Angle limit (

\theta_{max}

): 180°

Wedge center radius: 2.0 m

Wedge center angle: 45°

Double Integral Formulation

\int_0^{\theta_{max}}\int_0^{R_{max}} r\,dr\,d\theta

Finite wedge area:0.1200 m²

Analytical integral:14.1372 m²

Polar sweeping grid

Wedge bounds: $1.80\le r\le2.20$ and $0.71\le\theta\le0.86$ .

The polar area element

In Cartesian coordinates, $dA=dx\\,dy$ is locally rectangular.

In polar coordinates, radial thickness is $dr$ and angular arc length is $r\\,d\theta$ .

dA=r\\,dr\\,d\theta

Moving the red wedge outward increases its area while $dr$ and $d\theta$ remain fixed. This illustrates the geometric scaling represented by the Jacobian factor $r$ .

The current preset reports an analytical integral for the entire purple sector, so the displayed value can be checked independently of the wedge visualization.

Applications to Area and Volume

Area of a Planar Region

Integrate one over the region.

A=\iint_R 1\,dA.

Signed Volume Under a Graph

Signed accumulation between z=f(x,y) and the xy-plane.

V_{\mathrm{signed}}=\iint_R f(x,y)\,dA.

If $f\ge 0$ on $R$ , this signed value is also the geometric volume under the surface. If $f$ changes sign, geometric volume requires handling $|f|$ or splitting the domain.

Mass, First Moments, and Centroids of Laminas

For a thin lamina occupying $R$ with surface mass density $\rho(x,y)\ge 0$ , density has units of mass per area. Integrating density over area gives mass.

Lamina Mass

Total mass of a variable-density lamina.

m=\iint_R \rho(x,y)\,dA.

First Moments

Moments about the coordinate axes.

M_x=\iint_R y\,\rho(x,y)\,dA, \qquad M_y=\iint_R x\,\rho(x,y)\,dA.

Center of Mass

Density-weighted average position.

\bar{x}=\frac{M_y}{m}, \qquad \bar{y}=\frac{M_x}{m}.

$Moment naming$

$M_y$ contains the lever arm $x$ because it is the moment about the y-axis. Likewise, $M_x$ contains the lever arm $y$ .

Planar Moments of Inertia

Lamina Moments of Inertia

Resistance of a lamina to angular acceleration about coordinate axes.

I_x=\iint_R y^2\rho\,dA, \qquad I_y=\iint_R x^2\rho\,dA, \qquad I_O=I_x+I_y=\iint_R (x^2+y^2)\rho\,dA.

Introductory Surface Area

A smooth surface can be described parametrically by a vector-valued map $\mathbf{r}(u,v)$ . The tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ span a small parallelogram on the surface, so the magnitude of their cross product gives the local area-scaling factor.

Parametric Surface Area

Area of a smooth parametrized surface over a parameter region D.

S = \iint_D \left\lVert \mathbf{r}_u\times\mathbf{r}_v \right\rVert \,du\,dv.

For a smooth graph $z=f(x,y)$ above a planar region $R$ , use the parameterization $\mathbf{r}(x,y)=\langle x,y,f(x,y)\rangle$ . The general formula then reduces to the familiar graph formula.

Surface Area of z=f(x,y)

Graph form of the parametric surface-area formula.

S=\iint_R \sqrt{1+f_x^2+f_y^2}\,dA.

$Surface area is not signed volume$

The norm $\lVert\mathbf{r}_u\times\mathbf{r}_v\rVert$ is nonnegative. Surface area measures geometric area, whereas $\iint_R f\,dA$ may be signed when $f$ changes sign.

Key Takeaways

A double integral is a signed accumulation over a two-dimensional region.
Fubini's Theorem converts a double integral into iterated single-variable integrals.
Type I and Type II descriptions encode the same region using different slice directions; changing order requires rebuilding the bounds.
Reversed limits change orientation and therefore change sign.
Polar coordinates use $dA=r\,dr\,d\theta$ ; the factor $r$ is essential.
Double integrals compute planar area, volume under nonnegative surfaces, lamina mass, centroids, planar moments of inertia, and surface area through a two-parameter description.
Parametric surface area uses $\lVert\mathbf{r}_u\times\mathbf{r}_v\rVert$ as the local area scale; the graph formula is a special case.
Topic 07 extends multiple integration to three-dimensional solids and the full change-of-variables theorem.

Previous TopicApplications of Integration - Worked Examples

Quiz Me

Next TopicMultiple Integrals - Examples & Applications

Prev Next

Quiz Me