Question

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer. $$\int_{y=0}^{2} \int_{x=2 y}^{4} d x d y$$

Short answer

The value of the integral is 4.

Step-by-step solution

Identify the limits of integration

First, note the limits of integration. The outer integral is over variable \(y\) from 0 to 2. The inner integral is over variable \(x\) from \(2y\) to 4.

Write the inner integral

Integrate the inner integral with respect to \(x\): \[ \int_{2y}^{4} dx \]

Evaluate the inner integral

The integral of 1 with respect to \(x\) is \(x\). So, evaluate \( \int_{2y}^{4} dx \): \[ x \bigg|_{2y}^{4} = 4 - 2y \]

Write and evaluate the outer integral

Substitute the result from the inner integral into the outer integral: \[ \int_{0}^{2} (4 - 2y) dy \] Now integrate with respect to \(y\).

Integrate the outer integral

Find the integral of \(4 - 2y\) with respect to \(y\): \[ \int_{0}^{2} (4 - 2y) dy = 4y - y^2 \bigg|_{0}^{2} \]

Evaluate the result

Substitute the limits into the result: \[ (4(2) - (2)^2) - (4(0) - (0)^2) = 8 - 4 - 0 = 4 \]

Key concepts

limits of integration

In double integrals, setting up the correct limits of integration is crucial. The limits define the region over which you're integrating. There are two sets of limits: one for the outer integral and one for the inner integral.
In our problem, the outer integral for variable \( y \) ranges from 0 to 2, while the inner integral for variable \( x \) ranges from \( 2y \) to 4. This nesting of integrals allows us to integrate step-by-step, leading us to the final solution.

evaluating integrals

To evaluate double integrals, follow the nested structure of inner and outer integrals. Start with the inner integral, integrate it, and then use the result in the outer integral.
Integrating is a two-step process where you:

• Integrate the inner integral while treating the outer variable as a constant.
• Use the result in the outer integral and integrate it.

Following this process step-by-step ensures you evaluate the integral accurately, leading to the correct answer.

inner integral

The inner integral is the first part of the nested integration process. You integrate with respect to the inner variable while treating the outer variable as a constant.
In our problem, the inner integral is \[ \int_{2y}^{4} dx \].
Since the integrand is 1, the integral of 1 with respect to \( x \) is simply \( x \). Hence, we have:
\[ x \bigg|_{2y}^{4} \]
Evaluating this from \( 2y \) to 4 gives \( 4 - 2y \). This result will be used in the outer integral.

outer integral

After evaluating the inner integral, substitute its result into the outer integral to continue the process. The outer integral is integrated with respect to its variable.
For our problem, the outer integral becomes:
\[ \int_{0}^{2} (4 - 2y) dy \].
We now integrate \( 4 - 2y \) with respect to \( y \), yielding:
\[ 4y - y^2 \bigg|_{0}^{2} \].
Evaluating this from 0 to 2 gives \( (8 - 4) - (0 - 0) = 4 \).

integration techniques

Using proper integration techniques simplifies the process, ensuring you'll arrive at the correct solution. For double integrals:

• Focus first on your inner integral, treating outer variables as constants.
• Carefully address the limits of integration.
• Handle one integral at a time, then substitute and integrate the result.

This methodical approach ensures you handle each step systematically and accurately.