Double Integrals

In mathematics, double integral is defined as the integrals of a function in two variables over a region in $R^2$, i.e. the real number plane. The double integral of a function of two variables, say $f(x, y)$ over a rectangular region can be denoted as:

$$\iint_R f(x, y) \, dA$$

where $dA$ represents an infinitesimal area element in R. There are two common ways to express and evaluate a double integral: as iterated integrals and by using a specific order of integration.

Evaluating Double Integrals

Iterated Integrals

To evaluate a double integral, we typically express it as an iterated integral. This involves integrating first with respect to one variable and then with respect to the other. The order of integration can be $dx$ first and then $dy$, or $dy$ first and then $dx$, depending on the region $R$.

For example, if $R$ is defined by $a \leq x \leq b$ and $g_1(x) \leq y \leq g_2(x)$, the double integral is:

if $R$ is defined by $a \leq x \leq b$ and $g_1(x) \leq y \leq g_2(x)$, the double integral is:

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

If $R$ is defined by $c \leq y \leq d$ and $h_1(y) \leq x \leq h_2(y)$, the double integral is:

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

Change of Order of Integration

Steps for Changing the Order of Integration:

1. Sketch the Region of Integration: Visualize the region R over which the integration is performed.

2. Determine the Bounds for the New Order: Express the region R in terms of the new order of integration.

3. Rewrite the Integral: Change the order of the integral and adjust the limits accordingly.

Polar Coordinates

Polar coordinates represent a point in a plane using a distance from a reference point and an angle from a reference direction. This system is particularly useful in contexts where the relationship between points is more naturally expressed in terms of angles and distances, such as in circular and rotational systems.

Components of Polar Coordinates:

1. Radial Distance (r): The distance from the origin (also called the pole) to the point.

2. Angular Coordinate (θ): The angle measured from the positive x-axis to the line connecting the origin to the point. The angle is typically measured in radians or degrees.

   A point P in polar coordinates is represented as (r,θ).

Graphing of Polar Curves

Polar curves are graphs of equations expressed in polar coordinates, typically in the form $r=f(θ)$, where r is the radius and θ is the angle. Below are the steps and examples for graphing polar curves.

Steps to Graph Polar Curves:

1. Identify the Equation: Determine the form of the polar equation $r=f(θ)$.
2. Create a Table of Values: Choose values for θ (typically ranging from 0 to $2π$ or 0 to $360^∘$) and calculate the corresponding r values
3. Plot Points: Plot each $(r,θ)$ point on polar graph paper
4. Connect Points Smoothly: Draw a smooth curve through the plotted points to represent the polar curve.

Change of variables

Converting from Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$ involves expressing the position of a point in terms of its distance from the origin and its angle from the positive x-axis.

Conversion Formulas:

1. From Cartesian to Polar:

   • Radial Distance r : $$r = \sqrt{x^2 + y^2}$$
   • Angle $\theta$ (measured from the positive x-axis): $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ Note: The angle $\theta$ may need adjustment based on the quadrant in which the point $(x, y)$ lies.

2. From Polar to Cartesian:

   • x :
   $$\;\;\; \; \; x = r \cos(\theta)$$
   • y :
   $$\;\;\; \; \; y = r \sin(\theta)$$

Example: Conversion from Cartesian to Polar Coordinates:

Ques. Convert the Cartesian coordinates $(3, 4)$ to polar coordinates.

1. Calculate r:

   $r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2. Calculate $\theta$:

   $$\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{4}{3}\right)$$

   Using a calculator to find $\tan^{-1}(4/3)$:

   $$\theta \approx 53.13^\circ$$

So, the polar coordinates are $(5, 53.13^\circ)$.

Applications of Double Integrals to Areas and Volumes

Double integrals are a powerful tool in calculus for computing areas and volumes in two and three dimensions.

Area of a Region

The double integral can be used to find the area of a region R in the xy-plane. If R is a region bounded by the curves $x = a$, $x = b$, $y = c$, and $y = d$, then the area A of R is given by:

$$A = \iint_R 1 \, dA$$

In Cartesian coordinates, this becomes:

$$A = \int_a^b \int_c^d 1 \, dy \, dx$$

Volume Under a Surface

The double integral can also be used to find the volume under a surface $z = f(x, y)$ over a region R in the xy-plane. The volume V is given by:

$$V = \iint_R f(x, y) \, dA$$

In Cartesian coordinates, this becomes:

$$V = \int_a^b \int_c^d f(x, y) \, dy \, dx$$

Evaluation of Triple Integral

Triple integrals extend the concept of double integrals to three dimensions, allowing us to calculate volumes, masses, and other quantities over three-dimensional regions. To evaluate a triple integral, we integrate a function of three variables over a three-dimensional region.

The general form of a triple integral is:

$$\iiint_R f(x, y, z) \, dV$$

where R is the region of integration and dV represents the differential volume element, which in Cartesian coordinates is $dx \, dy \, dz$.

Step-by-Step Evaluation

To evaluate a triple integral, follow these steps:

1. Define the region of integration: Determine the limits for x, y, and z
2. Set up the integral: Write the integral with the appropriate limits and the integrand $f(x, y, z)$
3. Integrate iteratively: Perform the integration in the order specified, typically starting with the innermost integral and working outward.