Problem 34 - Calculus an Applied Approach

Question

Evaluate the double integral. Note that it is necessary to change the order of integration. $$ \int_{0}^{2} \int_{x}^{2} e^{-y^{2}} d y d x $$

Step-by-Step Solution

Answer

So, the value of the given double integral after changing the order of integration is \(\sqrt{\pi}(1 + 2e^{-2})/2\).

Step 1: Visualize the Limits of Integral

Start by understanding the limits of integration given in the problem. For the given double integral, the x-variable is from 0 to 2, and y-variable is from x to 2. Here, y is dependent on x and takes value greater than or equal to x and less than or equal to 2, which explains why we need to change the order of integration.

Step 2: Change the Order of Integration

To change the order of integration, note that y runs from x to 2. So to reverse this, x will run from 0 to y. And y will run from 0 to 2 as x runs from 0 to 2 for our original equation. So we can write our integral as follows: \[ \int_{0}^{2} \int_{0}^{y} e^{-y^{2}} dx dy \].

Step 3: Evaluate the Inner Integral

The inner integral has the respect to x, but it has no x in the integration. So, the solution of the inner integral will be x times the integral, from 0 to y. We then have \[ \int_{0}^{2} y e^{-y^{2}} dy \].

Step 4: Solve the Integral

Now, the integral is just a standard one-dimensional integral, but it has to be solved using substitution. Let \(u = -y^{2}\), then \(du = -2y dy\), or \(dy = -du / (2y) = -du / (2\sqrt{-u})\). The limits change with the substitution, when \(y = 0\), \(u = 0\), when \(y = 2\), \(u = -4\). So our integral becomes \[ -1/2 \int_{0}^{-4} e^{u} du / \sqrt{-u} \]. After the change of variables the bounds must be adjusted. The new bounds are 0 to -4. Since the integral is improper at 0, separate it into two parts: \[ -1/2 \int_{-1}^{-4} e^{u} du / \sqrt{-u} -1/2 \int_{0}^{-1} e^{u} du / \sqrt{-u} \].

Step 5: Finalize the Solution

Both integrals can be solved using the method of integration by parts. After solving, the answer is \( (1 - e^{-4})(2\sqrt{\pi})/4 + \sqrt{\pi}/2 = \sqrt{\pi}/2 + e^{-2}\sqrt{\pi}\) = \(\sqrt{\pi}(1 + 2e^{-2})/2\).

Key Concepts

Integration Techniques

Integration techniques are essential tools that help solve integrals that aren't straightforward. In double integrals, like in our problem, these techniques can involve multiple steps to get to the solution.

Most often, you'll deal with techniques like:

• Substitution: Useful when you can replace expressions to simplify the integral.
• Integration by parts: Helpful when the integral is a product of functions.
• Changing the order of integration: Often necessary in double integrals for easier computation.

When you approach these problems, always look for patterns or transformations that could simplify the integral. This step might require a bit of creativity and practice, but as you become more familiar with these techniques, the problem-solving process will become more intuitive.

Order of Integration

In some double integrals, as seen in our problem, changing the order of integration makes solving the integral significantly simpler. This means you swap the positions of the integrals, making the outer integral become the inner and vice versa.

In the original setup, you have limits based on another variable, like having the y-variable range depend on x. To change this:

• Identify the new range for the inner integral.
• Keep consistent with the original limits but rearrange them.
• Update the integration limits correspondingly, aligning with the incremental shifts for the integral to make sense.

This change sometimes uncovers a simpler dependency or allows a more straightforward integration, making the integral easier to visualize and solve.

Substitution Method

The substitution method simplifies integrals by converting them into a form that's easier to work with. For example, in our exercise, substitution was used at the step of solving the single integral:

- Start by identifying a function within the integral, which when replaced, expresses the integral more simply. - Define a new variable, typically noted as u, and express its derivative. - Change the limits of integration according to the new variables involved.
This often turns the integral into a simple standard form that can be solved more easily. In the provided exercise, by substituting with the expression related to y, we were able to transition the integration to a uniform and familiar territory.

Integration by Parts

Integration by parts is a technique based on decomposing an integral of products into simpler parts. This is especially useful when dealing with double integrals that boil down to standard integrals after some manipulation. The method is derived from the product rule of differentiation and can be understood as:
If you have an integral in the form of: \[ \int u \, dv = uv - \int v \, du \]
For our problem, this technique was used to deal with individual simpler parts after substitution was done.

• Identify which part of the integral to differentiate (u) and which to integrate (dv).
• Differentiate u to get du, and integrate dv to find v.
• Substitute back into the formula and simplify.

By repeating these steps, integrals that look complicated turn into manageable ones, facilitating finding the solution, much like in the example provided.