Question

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x y d A\\\ &R \text { : rectangle with vertices at }(0,0),(0,5),(3,5),(3,0) \end{aligned} $$

Short answer

The value of the double integral over the region R is \(\frac{225}{4}\).

Step-by-step solution

Setup the integral in dx dy order

In this case, integrate with respect to x first, then y. So, as the limits of x are from 0 to 3 and for y are from 0 to 5, the double integral will be as follows: \(\int_0^5 \int_0^3 xy \, dx \, dy\).

Setup the integral in dy dx order

Now, let's setup the double integral in the reverse order, where we integrate with respect to y first, then x. The limits for x and y remain the same, so we get: \(\int_0^3 \int_0^5 xy \, dy \, dx\).

Evaluate the more convenient integral

In this case, both the integrals are equivalent because the limits are constants and the integrand depends on both x and y, thus either integral can be evaluated conveniently. Here, we will evaluate \(\int_0^5 \int_0^3 xy \, dx \, dy\). First, we integrate \(xy\) with respect to x from 0 to 3 with y held constant, which gives \( \frac{3^2y}{2} - 0 \). Then we integrate \(\frac{9y}{2}\) with respect to y from 0 to 5, and the integral evaluates to \(\frac{9}{2} \times \frac{5^2}{2} - 0 = \frac{225}{4}\).

Key concepts

Order of Integration

When dealing with double integrals, understanding the order of integration is crucial. The order of integration refers to the sequence in which you perform the integration with respect to each variable. For example, in the provided exercise, we have two possible orders:

• First integrate with respect to x, then with respect to y (\( \int_0^5 \int_0^3 xy \, dx \, dy \)
• First integrate with respect to y, then with respect to x (\( \int_0^3 \int_0^5 xy \, dy \, dx \)

Each order of integration may offer different levels of complexity. The choice often depends on the limits of integration or the integrand's complexity.

In this particular exercise, the integrals are straightforward due to constant limits and a simple integrand. Therefore, either order can be convenient to use.

Limits of Integration

Limits of integration are essential for defining the region over which integration is performed. In the given exercise, the region is a rectangle with vertices at \((0,0),(0,5),(3,5),(3,0)\). This indicates that:

• The variable x varies from 0 to 3
• The variable y varies from 0 to 5

These limits remain consistent regardless of the order of integration.

When setting up the double integral, ensure that these limits accurately represent the region being integrated over. For rectangular regions, as with this problem, the limits are constants. However, for other regions, such as triangles or circles, these limits may vary depending on the variable considered.

Evaluating Integrals

Evaluating double integrals involves solving two separate integrals sequentially according to the determined order. Here, we chose to evaluate \( \int_0^5 \int_0^3 xy \, dx \, dy \).

First, integrate with respect to x:

• Fix y and integrate \(xy\) with limits from 0 to 3, resulting in \( \frac{9y}{2} \).

Next, integrate the result with respect to y:

• Integrate \( \frac{9y}{2} \) from 0 to 5, which equals \( \frac{225}{4} \).

Each of these steps simplifies the problem further, breaking it into manageable pieces. Regardless of the complexity, following a systematic process of first selecting your limits and then integrating step by step makes solving double integrals more approachable.