Question

Sketch the region $R$ whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $${\int }_0^1{\int }_0^{2}dydx$$

Short answer

The area of the region $R$ is 2 square units, showing that changing the order of integration in double integrals does not affect the computed area.

Step-by-step solution

Sketch the region R

The region $R$ is defined by the limits of the integral, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2. This describes an area in the xy-plane, being a rectangle with a length of 1 along the x-axis and a length of 2 along the y-axis.

Compute Area using the initial order of integration

Perform the inner integral first, keeping $x$ fixed as the outer integral implies. Since there's no function to integrate, only differentials $dy$ and $dx$, consider the integrand as 1. Then the integral ${\int }_0^{2}dy$ equals to 2. The outer integral thus becomes ${\int }_0^{1}2dx$, which equals to 2. The area of region $R$ is therefore 2 square units.

Change the order of integration

Let's change the order of integration to $dydx$, which results in the integral ${\int }_0^2{\int }_0^{1}dxdy$

Compute Area using the reversed order of integration

Perform the inner integral first, keeping $y$ fixed as the outer integral implies. Since there's still no function to integrate, only differentials $dx$ and $dy$, consider the integrand as 1. Then the integral ${\int }_0^{1}dx$ equals to 1. The outer integral thus becomes ${\int }_0^{2}1dy$, which equals to 2. The area of region $R$ is again found to be 2 square units, confirming that the area is unaffected by the order of integration.

Comparison

Compare the two areas obtained in steps 2 and 4. Since both cases resulted in the same area, it has been shown that the order of performing the integration in a double integral does not affect the computed area of the region.

Key concepts

Order of Integration

In double integrals, the order of integration refers to the sequence in which the integration is performed over the variables. In the context of a Cartesian plane, this usually pertains to the variables x and y. For instance, the double integral ${\int }_0^1{\int }_0^{2}dydx$ indicates that the integration with respect to y is to be performed first, followed by the integration with respect to x.

However, for certain regions, notably rectangular or box-type regions, the order of integration can be reversed without affecting the final result. This is what the exercise demonstrates by changing the integral to ${\int }_0^2{\int }_0^{1}dxdy$. This interchangeability is because the region is the same, and simple integration only accumulates the area across the bounded region without considering the path taken.

It's important for students to understand that although the order of integration can be flexible in such cases, it's not universally exchangeable. For more complex regions or integrands, changing the order might not be straightforward and could involve more detailed region sketching and setting up the limits of integration, a concept we'll explore in the next section.

Region Sketching in Calculus

Sketching a region in calculus, particularly when dealing with double integrals, is an essential skill. It allows students to visualize the area over which the integration takes place. For our given exercise, the region R is described by the double integral boundaries. The first integral goes from 0 to 1, which corresponds to the x-axis limits, and the second integral goes from 0 to 2, for the y-axis limits.

As we're working with a rectangle, sketching is simple: it's a matter of drawing a rectangle in the xy-plane with the given dimensions. However, for more complicated shapes, understanding the relationship between the integrals' limits and the shapes they represent becomes more challenging. For those cases, breaking the area down into simpler shapes can be beneficial. This sketching step is crucial before setting up integrals, as it ensures the correct limits are being used, whether one keeps the original order or changes it.

Integral Computation

The integral computation in the context of our double integral problem, involves two significant steps: evaluating the inner integral and then proceeding to the outer integral. As the region in our current exercise is straightforward, with the inner integral being ${\int }_0^{2}dy$ and the outer integral being ${\int }_0^{1}dx$, the computation is simple: you are effectively summing up the 'slices' of area across the shape described by R.

When dropping in the integrals into the computation, for integrals with more than differentials, one must involve the fundamental theorem of calculus to evaluate the definite integrals. This could get more involved when the function within the integral showcases x and y dependencies, but in our case, the integrand is 1, making the integral akin to summing uniform slices of area.

xy-Plane Area Calculation

When it comes to xy-plane area calculation using double integrals, the process is a reliable method to find the area of more complex shapes that cannot be handled with basic geometry formulas. The beauty of double integrals lies in their ability to accumulate the area slice by slice, or in terms of calculus, 'element by element'.

This integral, ${\int }_0^1{\int }_0^{2}dydx$, simplifies to an area computation because the integrand is 1, which can be interpreted as summing unit areas over the specified region, R. This is a cornerstone concept in multivariable calculus that extends to more complex scenarios involving curves and variable integrands, allowing for the area under surfaces to be calculated with precision.