Unit 5 - Notes

MTH165

Unit 5: Multiple Integrals

1. Double Integrals

1.1 Definition

A double integral is the generalization of the definite integral to functions of two variables. Geometrically, if $f(x,y) \geq 0$ , the double integral represents the volume of the solid that lies above the region $R$ in the $xy$ -plane and below the surface $z = f(x,y)$ .

Notation:

\iint_R f(x, y) \, dA

Where

dA

represents an infinitesimal area element (

dx\,dy

dy\,dx

1.2 Evaluation using Iterated Integrals

Double integrals are evaluated by computing two single integrals successively. The order of integration depends on the definition of the region $R$ .

Case 1: Region bounded by Vertical Lines (Type I)
If the region $R$ is defined by $a \leq x \leq b$ and $g_1(x) \leq y \leq g_2(x)$ :

\iint_R f(x, y) \, dx \, dy = \int_a^b \left[ \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \right] dx

Integrate with respect to $y$ first (treat $x$ as constant).
Integrate the result with respect to $x$ .

Case 2: Region bounded by Horizontal Lines (Type II)
If the region $R$ is defined by $c \leq y \leq d$ and $h_1(y) \leq x \leq h_2(y)$ :

\iint_R f(x, y) \, dx \, dy = \int_c^d \left[ \int_{h_1(y)}^{h_2(y)} f(x, y) \, dx \right] dy

Integrate with respect to $x$ first (treat $y$ as constant).
Integrate the result with respect to $y$ .

2. Change of Order of Integration

2.1 Concept

Sometimes, evaluating an iterated integral in the given order (e.g., $dy\,dx$ ) is difficult or impossible analytically (e.g., $\int e^{x^2} dx$ ). Changing the order to $dx\,dy$ may simplify the integration.

2.2 Algorithm for Changing Order

Identify Limits: Write down the given inequalities for the region $R$ based on the current limits of integration.
Sketch the Region: Draw the curves bounding the region in the $xy$ -plane. Identify the intersection points.
Change the Strip:
- If the original was a vertical strip ( $y$ limits depend on $x$ ), draw a horizontal strip ( $x$ limits depend on $y$ ).
- If the original was horizontal, draw a vertical strip.
Determine New Limits:
- Find the new inner limits (functions of the outer variable).
- Find the new outer limits (constant values).
Evaluate: Set up the new integral and solve.

3. Application of Double Integrals

3.1 Calculation of Area

If we set the function $f(x,y) = 1$ , the double integral calculates the area of the 2D region $R$ .

Formula:

\text{Area} = \iint_R 1 \, dA = \iint_R dx \, dy

3.2 Calculation of Volume

If $f(x,y)$ defines a surface $z = f(x,y)$ above a region $R$ in the $xy$ -plane, the volume $V$ of the solid cylinder bounded by the surface and the plane is:

Formula:

\text{Volume} = \iint_R f(x, y) \, dx \, dy

4. Change of Variables (Double Integrals)

To simplify integrals over circular, cardioid, or complex regions, we change the coordinate system (e.g., Cartesian to Polar).

4.1 The Jacobian

For a transformation $x = x(u,v)$ and $y = y(u,v)$ , the Jacobian $J$ is the determinant of the partial derivatives matrix:

J = \frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}

Transformation Formula:

\iint_R f(x, y) \, dx \, dy = \iint_{R'} f(x(u, v), y(u, v)) \, |J| \, du \, dv

4.2 Polar Coordinates

Used for circular regions.

Substitution: $x = r \cos \theta$ , $y = r \sin \theta$
Jacobian: $|J| = r$
Area Element: $dx \, dy \Rightarrow r \, dr \, d\theta$

Integral in Polar Form:

\int_{\theta=\alpha}^{\beta} \int_{r=r_1(\theta)}^{r_2(\theta)} f(r \cos \theta, r \sin \theta) \cdot r \, dr \, d\theta

5. Triple Integrals

5.1 Definition

A triple integral generalizes the concept to functions of three variables $f(x,y,z)$ over a solid region $D$ in 3D space.

Notation:

\iiint_D f(x, y, z) \, dV

Where

dV

is the volume element (

dx\,dy\,dz

5.2 Evaluation (Cartesian Coordinates)

Usually evaluated as iterated integrals:

\int_{x=a}^b \int_{y=g_1(x)}^{g_2(x)} \int_{z=h_1(x, y)}^{h_2(x, y)} f(x, y, z) \, dz \, dy \, dx

Inner Integral: Integrate w.r.t $z$ (limits are surfaces $z=h_1$ to $z=h_2$ ).
Middle Integral: Integrate w.r.t $y$ (limits are curves $y=g_1$ to $y=g_2$ ).
Outer Integral: Integrate w.r.t $x$ (limits are constants $x=a$ to $x=b$ ).

6. Change of Variables (Triple Integrals)

Standard coordinate systems used to simplify triple integrals involving cylinders or spheres.

6.1 Cylindrical Coordinates

Used for regions with symmetry about an axis (e.g., cylinders, cones).

Relationship: $x = r \cos \theta$ , $y = r \sin \theta$ , $z = z$
Fundamental: $x^2 + y^2 = r^2$
Jacobian: $|J| = r$
Volume Element: $dV = r \, dz \, dr \, d\theta$

Integral:

\iiint_D f(x, y, z) \, dV = \iiint_{D'} f(r \cos \theta, r \sin \theta, z) \cdot r \, dz \, dr \, d\theta

6.2 Spherical Coordinates

Used for regions with symmetry about a point (e.g., spheres, cones, portions of spheres).

Variables:
- $\rho$ (rho): Distance from origin to point ( $0 \leq \rho < \infty$ ).
- $\phi$ (phi): Angle from positive $z$ -axis ( $0 \leq \phi \leq \pi$ ).
- $\theta$ (theta): Angle in $xy$ -plane from positive $x$ -axis ( $0 \leq \theta \leq 2\pi$ ).
Relationship:
- $x = \rho \sin \phi \cos \theta$
- $y = \rho \sin \phi \sin \theta$
- $z = \rho \cos \phi$
Fundamental: $x^2 + y^2 + z^2 = \rho^2$
Jacobian: $|J| = \rho^2 \sin \phi$
Volume Element: $dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$

7. Application of Triple Integrals

7.1 Calculation of Volume

Just as the double integral of $f(x,y)=1$ gives Area, the triple integral of the function $f(x,y,z) = 1$ over a region $D$ calculates the volume of that region.

Formula:

\text{Volume} = \iiint_D 1 \, dV = \iiint_D dx \, dy \, dz

Summary of Volume Calculations

Method	Integral Setup	Usage Scenarios
Double Integral	$V = \iint_R z \, dx \, dy$	Used when calculating volume under a surface $z=f(x,y)$ bounded by a region $R$ in the $xy$ -plane.
Triple Integral	$V = \iiint_D 1 \, dx \, dy \, dz$	Used when the region $D$ is defined by complex 3D boundaries, or when utilizing cylindrical/spherical coordinates is necessary.

7.2 Dirichlet’s Integral (Special Case)

A useful formula for finding the volume of a tetrahedron bounded by coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ :

\iiint_V x^{l-1} y^{m-1} z^{n-1} \, dx \, dy \, dz = \frac{\Gamma(l)\Gamma(m)\Gamma(n)}{\Gamma(l+m+n+1)}

Where the region

V

x \ge 0, y \ge 0, z \ge 0

and

x+y+z \le 1

. (With scaling factors

a,b,c

applied if necessary).

Unit 5 - Notes

Table of Contents

Unit 5: Multiple Integrals

1. Double Integrals

1.1 Definition

1.2 Evaluation using Iterated Integrals

2. Change of Order of Integration

2.1 Concept

2.2 Algorithm for Changing Order

3. Application of Double Integrals

3.1 Calculation of Area

3.2 Calculation of Volume

4. Change of Variables (Double Integrals)

4.1 The Jacobian

4.2 Polar Coordinates

5. Triple Integrals

5.1 Definition

5.2 Evaluation (Cartesian Coordinates)

6. Change of Variables (Triple Integrals)

6.1 Cylindrical Coordinates

6.2 Spherical Coordinates

7. Application of Triple Integrals

7.1 Calculation of Volume

Summary of Volume Calculations

7.2 Dirichlet’s Integral (Special Case)