Question

\(\int_{0}^{1} \int_{5}^{\sqrt{x}}\left(x^{2}+y^{2}\right) d x d y=\)

Short answer

The integral is 0 because the region of integration is empty.

Step-by-step solution

Set Up the Integral

We are given a double integral \( \int_{0}^{1} \int_{5}^{\sqrt{x}} (x^2 + y^2) \; dx \; dy \). The region of integration is defined by \( y \) from 5 to \( \sqrt{x} \) and \( x \) from 0 to 1.

Check Feasibility of Limits

The inner integral requires \( y \) to go from 5 to \( \sqrt{x} \), within the bounds of \( x \) being from 0 to 1. However, since \( \sqrt{x} \) (given \( x \in [0,1] \)) will always be less than 5 for any \( x \) in the interval, the region of integration is empty.

Draw a Conclusion from the Empty Set

Since there is no real pair \((x, y)\) that satisfies \(5 \leq y \leq \sqrt{x}\) within \(x \in [0, 1]\), the region of integration does not exist. This means the integral evaluates to 0.

Key concepts

Region of Integration

When working with double integrals, understanding the region of integration is a critical first step. The region of integration is the area over which integration occurs, typically defined by the limits in the integral. In many math problems, it often corresponds to a specific area on an xy-plane.
When solving a double integral like \[\int_{0}^{1} \int_{5}^{\sqrt{x}} (x^2 + y^2)\, dx\, dy,\]we must first determine the range of values for both variables. This directly influences the area or 'region' over which we're integrating. In our example, this region is supposed to lie between the limits given as 0 to 1 for \(x\) and 5 to \(\sqrt{x}\) for \(y\). However, a closer look can sometimes lead us to surprising realizations about these regions. More on this is explained in the following sections.

Integral Limits

Integral limits are vital in setting the boundaries for integration. They specifically outline the extent of the region over which the function should be integrated. These limits can sometimes be actual numbers or, as in our example, can include expressions involving one of the variables such as \(y = \sqrt{x}\).
Consider the limit \(y = \sqrt{x}\) within the bounds of \(x\in[0,1]\). In a typical scenario, you would expect the integration to cover an area where all values align inside these bounds. However, in our example, before we start calculating, it's smart to check if these limits make sense together.
A careful analysis shows that \(\sqrt{x}\) ranges from 0 to 1 as \(x\) goes from 0 to 1, which is always less than 5. Hence, it reveals an interesting scenario where no \(y\) within the set limits actually exists, leading us directly into discussions about empty regions of integration.

Empty Region Evaluation

An essential, albeit sometimes overlooked, aspect of double integration is ensuring that the region of integration isn't empty. An empty region of integration means there are no real values within the specified limits that satisfy the conditions of the integral.
In our case, with \(5 \leq y \leq \sqrt{x}\), and knowing that \(\sqrt{x}\) for \(x\in[0,1]\) can never reach or exceed 5, we end up with an empty region. Consequently, there are no valid \((x, y)\) pairs to integrate over.
This means the integral simplifies drastically. Since integration is over an empty region, the total sum of all the infinitesimal areas effectively becomes zero. Thus, the integral evaluates simply to 0, saving us calculations but offering a great insight into the importance of verifying region feasibility before proceeding with any integration task.