Unit 6 - Notes

MTH174 5 min read

Unit 6: Integral Calculus

1. Double Integrals

A double integral is the extension of a definite integral to two dimensions. It is used to integrate a function of two variables, $f(x,y)$ , over a two-dimensional region $R$ in the $xy$ -plane.

Definition

The double integral of $f(x,y)$ over a region $R$ is denoted as:
$\iint_R f(x,y) \, dA$
where $dA$ represents an infinitesimal element of area ( $dA = dx \, dy$ or $dA = dy \, dx$ ).

Evaluation using Fubini's Theorem

Fubini's Theorem states that if $f(x,y)$ is continuous on a closed bounded region $R$ , the double integral can be evaluated as an iterated integral.

Type I Region (Vertical Strips):
If the region $R$ is bounded by $a \le x \le b$ and $g_1(x) \le y \le g_2(x)$ , the integral is evaluated as:
$\iint_R f(x,y) \, dA = \int_{a}^{b} \left( \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \right) dx$
Here, integrate with respect to $y$ first, treating $x$ as a constant.
Type II Region (Horizontal Strips):
If the region $R$ is bounded by $c \le y \le d$ and $h_1(y) \le x \le h_2(y)$ , the integral is:
$\iint_R f(x,y) \, dA = \int_{c}^{d} \left( \int_{h_1(y)}^{h_2(y)} f(x,y) \, dx \right) dy$
Here, integrate with respect to $x$ first, treating $y$ as a constant.

2. Change of Order of Integration

Sometimes, evaluating an iterated integral in the given order is difficult or impossible (e.g., integrating $\int e^{x^2} dx$ ). Reversing the order of integration can simplify the problem.

Steps to Change the Order

Identify the given limits: Extract the bounds for $x$ and $y$ from the given iterated integral.
Sketch the region of integration ( $R$ ): Draw the curves representing the boundaries of the region on the Cartesian plane.
Determine the new strips:
- If the original integral used vertical strips ( $dy \, dx$ ), switch to horizontal strips ( $dx \, dy$ ).
- If the original integral used horizontal strips ( $dx \, dy$ ), switch to vertical strips ( $dy \, dx$ ).
Find the new limits: Based on the new strips, determine the new inner limits (functions) and outer limits (constants).
Set up and evaluate the new integral.

3. Change of Variables in Double Integrals

When a region $R$ is difficult to describe in Cartesian coordinates (e.g., circles, ellipses), changing variables to a new coordinate system $(u, v)$ can vastly simplify the integral.

The Jacobian

If $x = g(u,v)$ and $y = h(u,v)$ , the Jacobian of the transformation is the determinant:
$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$

The change of variables formula is:
$\iint_R f(x,y) \, dx \, dy = \iint_S f(g(u,v), h(u,v)) |J| \, du \, dv$
where $S$ is the region in the $uv$ -plane corresponding to $R$ in the $xy$ -plane.

Polar Coordinates (Most Common Transformation)

For circular or radial regions, polar coordinates $(r, \theta)$ are ideal.

Transformations: $x = r \cos\theta$ , $y = r \sin\theta$
Jacobian: $|J| = r$
Differential Area: $dx \, dy = r \, dr \, d\theta$

Formula:
$\iint_R f(x,y) \, dA = \int_{\theta_1}^{\theta_2} \int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta, r\sin\theta) \, r \, dr \, d\theta$

4. Applications of Double Integrals

Calculating Area of a Plane Region

The area $A$ of a two-dimensional closed region $R$ can be found by setting the integrand $f(x,y) = 1$ .
$\text{Area} = \iint_R 1 \, dA = \iint_R dx \, dy$

Calculating Volume Under a Surface

The volume $V$ of a solid bounded above by the surface $z = f(x,y)$ (where $f(x,y) \ge 0$ ) and below by the region $R$ in the $xy$ -plane is:
$\text{Volume} = \iint_R f(x,y) \, dA$

5. Triple Integrals

Triple integrals extend the concept of double integrals to three dimensions, integrating a function of three variables $f(x,y,z)$ over a solid region $V$ in space.

Definition

$\iiint_V f(x,y,z) \, dV$
where $dV = dx \, dy \, dz$ (in Cartesian coordinates).

Evaluation

By extending Fubini's Theorem, a triple integral is evaluated as three iterated integrals. The most common order is integrating with respect to $z$ first:
$\iiint_V f(x,y,z) \, dV = \int_{a}^{b} \int_{g_1(x)}^{g_2(x)} \int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z) \, dz \, dy \, dx$

Inner integral ( $dz$ ): Evaluated from surface to surface (limits are functions of $x$ and $y$ ).
Middle integral ( $dy$ ): Evaluated from curve to curve in the projection region (limits are functions of $x$ ).
Outer integral ( $dx$ ): Evaluated from point to point (limits are constants).

6. Applications of Triple Integrals to Calculate Volume

Just as the double integral of $1$ gives area, the triple integral of $1$ gives the volume of a 3D solid.

Basic Volume Formula

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

Volume Using Coordinate Transformations

Often, calculating volume is easier in non-Cartesian coordinate systems. The change of variables principle applies, requiring the use of a 3D Jacobian.

1. Cylindrical Coordinates

Best used when the solid has an axis of symmetry (e.g., cylinders, paraboloids).

Transformations: $x = r \cos\theta$ , $y = r \sin\theta$ , $z = z$
Volume Element: $dV = r \, dz \, dr \, d\theta$
Volume Formula:
$V = \iiint_V r \, dz \, dr \, d\theta$

2. Spherical Coordinates

Best used when the solid has a point of symmetry (e.g., spheres, cones).

Variables: $\rho$ (distance from origin), $\phi$ (angle from positive z-axis), $\theta$ (angle in xy-plane).
Transformations:
$x = \rho \sin\phi \cos\theta$
$y = \rho \sin\phi \sin\theta$
$z = \rho \cos\phi$
Jacobian / Volume Element: $dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$
Volume Formula:
$V = \iiint_V \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$

Unit 5

Unit 6 - Notes

Table of Contents

Unit 6: Integral Calculus

1. Double Integrals

Definition

Evaluation using Fubini's Theorem

2. Change of Order of Integration

Steps to Change the Order

3. Change of Variables in Double Integrals

The Jacobian

Polar Coordinates (Most Common Transformation)

4. Applications of Double Integrals

Calculating Area of a Plane Region

Calculating Volume Under a Surface

5. Triple Integrals

Definition

Evaluation

6. Applications of Triple Integrals to Calculate Volume

Basic Volume Formula

Volume Using Coordinate Transformations

1. Cylindrical Coordinates

2. Spherical Coordinates