Question

Evaluate the double integral. Note that it is necessary to change the order of integration. $$ \int_{0}^{3} \int_{y}^{3} e^{x^{2}} d x d y $$

Short answer

The value of the double integral after changing the order of integration and computing is \(\frac{1}{2}e^{9} - \frac{1}{2}\).

Step-by-step solution

Interpret the Integral Limits

We have \(\int_{0}^{3} \int_{y}^{3} e^{x^{2}} dx dy\). The inner integral has limits \(y\leq x \leq 3\) and the outer integral has \(0 \leq y \leq 3\). The defined region for this double integral in the xy-plane is as such: for every fixed y in [0,3], x varies between y and 3.

Change the Order of Integration

To express the limits in terms of \(dx dy\), let's envisage the region differently. Now for every fixed x in [0,3], y will vary from 0 to x. So, the limits can be rewritten as \(\int_{0}^{3} \int_{0}^{x} e^{x^{2}} dy dx\).

Evaluate the Inner Integral

Next, compute the inner integral which is a easy one with respect to y: \(\int_{0}^{x} e^{x^{2}} dy\), which will simply be \(ye^{x^{2}}\) evaluated between 0 and x. This will give us \(x e^{x^{2}}\).

Evaluate the Outer Integral

Substitute the above result into the outer integral and then compute the outer integral with respect to x: \(\int_{0}^{3} x e^{x^{2}} dx\). We'd use substitution method to compute this integral where u = \(x^2\). After substitution and integrating, we get \(\frac{1}{2}e^{x^{2}}\) evaluated between 0 to 3, simplifying to \(\frac{1}{2}e^{9} - \frac{1}{2}\).

Key concepts

Order of Integration

When evaluating double integrals, it's crucial to understand the order of integration. This order determines how we integrate over a region. Suppose we have the integral \( \int_{0}^{3} \int_{y}^{3} e^{x^{2}} dx \, dy \). Here, the order of integration starts with \(dx\) followed by \(dy\). This means we are first integrating with respect to \(x\) while keeping \(y\) constant, and then we integrate the result with respect to \(y\).

However, it can sometimes be more convenient to change the order of integration. This may simplify the integral or make it possible to evaluate if the original order is too complex. To change the order, we must re-examine the integration limits. Start by drawing the region of integration on a coordinate plane. For the given integral, the region describes a triangular area bounded by \(y = x\), \(x = 3\) and \(y = 0\). By changing the order, the limits become \(\int_{0}^{3} \int_{0}^{x} e^{x^{2}} dy \, dx \).
Why change the order? Sometimes changing the order of integration is necessary to evaluate the integral, especially if the integrand or region is complex. It makes the process more straightforward.

Definite Integrals

Definite integrals are used to find the actual value of the integral over a specified interval. In the context of double integrals, they describe the accumulation of quantities over a two-dimensional region. For example, the given problem involves evaluating \(\int_{0}^{3} \int_{0}^{x} e^{x^{2}} dy \, dx\) over a triangular region in the \(xy\)-plane.

Definite integrals have fixed limits, indicating the start and end points of integration. For the inner integral with respect to \(y\), the limits are from 0 to \(x\), while for the outer integral with respect to \(x\), the limits are from 0 to 3. Evaluating these integrals step-by-step gives us the final accumulation value.
It's important to interpret these limits correctly. The region described by a double integral provides the foundation for understanding how and why the function behaves over these intervals. It also helps in visualizing the bounded area, which is significant when solving integrals in real-world applications.

Substitution Method

The substitution method is a powerful tool in calculus for simplifying integrals. It involves replacing a part of the integral with a new variable to make the integration process easier. In the context of the original problem, we encounter this method when evaluating \( \int_{0}^{3} x \, e^{x^2} dx \).

To use substitution, set a new variable \(u = x^2\). Consequently, the differential \(du\) is \(2x \, dx\), or rearranged, \(dx = \frac{du}{2x}\). This substitution simplifies the integral into a more manageable form:

• Original integral: \(\int x \, e^{x^2} dx\)
• Substitution: Let \(u = x^2\), \(du = 2x \, dx\)
• Rewriting: \(\frac{1}{2} \int e^u du\)

By integrating \(e^u\), we arrive at \(\frac{1}{2} e^{x^2}\). After evaluating from 0 to 3, we find \(\frac{1}{2} e^9 - \frac{1}{2}\).
The substitution method is particularly helpful for integrals that involve complex expressions. It changes the perspective of the integral, thus making the calculation process simpler and more intuitive. Understanding when and how to use substitution is key to solving challenging integrals effectively.