What are Double Integrals over Rectangles in Mathematics?
Double integrals are a way to extend the concept of an integral to functions of two variables. When we integrate a function of two variables over a rectangular region in the xy-plane, we are computing the double integral over a rectangle.
What is the Rectangular Region?
A rectangular region is defined by two intervals: one for the x-variable and one for the y-variable. Suppose the intervals are [a, b] for x and [c, d] for y. The rectangular region can be written as:
R = {(x, y) | a ? x ? b, c ? y ? d}
How Do We Define the Double Integral Over a Rectangle?
The double integral of a function f(x, y) over a rectangular region R is written as:
?_R f(x, y) dA
Here, dA represents the differential area element in the xy-plane. For rectangular coordinates, this can be written as dA = dx dy (or equivalently dy dx, as the order of integration can often be interchanged under certain conditions).
What is the Process for Computing a Double Integral Over a Rectangle?
To compute the double integral, follow these steps:1. Set Up the Integral: Express the double integral in terms of iterated integrals. Write it as: ? from c to d [? from a to b f(x, y) dx] dy or ? from a to b [? from c to d f(x, y) dy] dx2. Evaluate the Inner Integral First: Choose an order of integration (either x-first or y-first), and compute the inner integral with respect to the chosen variable while treating the other variable as a constant.3. Evaluate the Outer Integral: After computing the inner integral, you will have a single-variable integral left. Evaluate this outer integral to obtain the final result.
Example to Illustrate the Process:
Consider the function f(x, y) = x + y over the rectangle R defined by 0 ? x ? 2 and 1 ? y ? 3.
1. Set Up the Integral: ? from 1 to 3 [? from 0 to 2 (x + y) dx] dy
2. Evaluate the Inner Integral: Compute ? from 0 to 2 (x + y) dx Since y is treated as constant regarding x, we get: ? from 0 to 2 (x + y) dx = ? from 0 to 2 x dx + ? from 0 to 2 y dx
- For ? from 0 to 2 x dx: (1/2)x^2 evaluated from x = 0 to x = 2 = (1/2)(2^2) - (1/2)(0^2) = 2
- For ? from 0 to 2 y dx: y * ? from 0 to 2 dx = y * [x evaluated from 0 to 2] = y * (2 - 0) = 2y
Therefore, ? from 0 to 2 (x + y) dx = 2 + 2y.
3. Evaluate the Outer Integral: Now, compute ? from 1 to 3 (2 + 2y) dy = ? from 1 to 3 2 dy + ? from 1 to 3 2y dy
- For ? from 1 to 3 2 dy: 2y evaluated from y = 1 to y = 3 = 2(3) - 2(1) = 6 - 2 = 4
- For ? from 1 to 3 2y dy: y^2 evaluated from y = 1 to y = 3 = [3^2 - 1^2] = 9 - 1 = 8
Therefore, ? from 1 to 3 (2 + 2y) dy = 4 + 8 = 12.
Hence, ?_R (x + y) dA = 12.
SummaryDouble integrals over rectangles involve integrating a function of two variables over a defined rectangular region. The process entails evaluating the inner and outer integrals iteratively, adjusting the order of integration according to the setup of the function and the given region.
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