Question

Evaluate the following integrals. $$\iint_{R} x \sec ^{2} y d A ; R=\left\\{(x, y): 0 \leq y \leq x^{2}, 0 \leq x \leq \sqrt{\pi} / 2\right\\}$$

Short answer

Based on the given step-by-step solution, create a short answer to the question: The value of the double integral $$\iint_R x \sec^2 y\,dA$$ where R is defined by \(0 \leq y \leq x^2\) and \(0 \leq x \leq \sqrt{\pi} / 2\) is given by $$\int_{0}^{\sqrt{\pi} / 2} x \tan(x^2) dx$$. However, this integral does not have an elementary function and requires the use of numerical techniques or series representations for evaluation.

Step-by-step solution

Setup the iterated integral

We are given the double integral of \(x\sec^2 y\) over the region, R. We will convert it into an iterated integral by integrating with respect to y first and then with respect to x. Recall that our region is defined by \(0 \leq y \leq x^2\) and \(0 \leq x \leq \sqrt{\pi} / 2\). Thus the integral can be written as: $$\int_{0}^{\sqrt{\pi} / 2} \int_{0}^{x^2} x \sec ^{2} y \, dy\, dx$$

Integrate with respect to y

First, we integrate with respect to y: $$\int_{0}^{\sqrt{\pi} / 2} \left[\int_{0}^{x^2} x \sec ^{2} y \, dy\right] dx$$ Since x is a constant with respect to y, we can take it outside the inner integration: $$\int_{0}^{\sqrt{\pi} / 2} x \left[\int_{0}^{x^2} \sec ^{2} y \, dy\right] dx$$ Now, we integrate \(\sec^2 y\) with respect to y. The integral of \(\sec^2 y\) is \(\tan y\): $$\int_{0}^{\sqrt{\pi} / 2} x \left[ \tan y\right]_0^{x^2} dx$$ Evaluate the limits: $$\int_{0}^{\sqrt{\pi} / 2} x (\tan(x^2) - \tan(0)) dx$$ Since \(\tan(0)=0\), the above expression simplifies to: $$\int_{0}^{\sqrt{\pi} / 2} x \tan(x^2) dx$$

Integrate with respect to x

Now, we need to integrate with respect to x: $$\int_{0}^{\sqrt{\pi} / 2} x \tan(x^2) dx$$ Unfortunately, we cannot find an elementary function for this integral. This means that we have to resort to other methods, such as using numerical techniques (e.g. Simpson's rule or the trapezoidal rule) or representations as series, to evaluate it. That being said, the result is often denoted using integral notation as a final answer: $$\int_{0}^{\sqrt{\pi} / 2} x \tan(x^2) dx$$

Key concepts

Double Integrals

A double integral extends the concept of a single integral to functions of two variables. Unlike regular integration, which deals with areas under curves on a single line, double integrals involve summing up a "volume" beneath a surface over a two-dimensional region. This might sound daunting, but it's a powerful tool for calculating the area, volume, and other quantities associated with the region.
Imagine you have a bumpy surface. The double integral helps find the "total weight" or "volume" if you were to stack blocks under the curve until it is flush with the surface.
Here's a step-by-step idea of how it works:

• First, define the region of integration, which is the area over which you'll apply the integral.
• Next, set up the double integral, respecting any limits given for the variables involved.
• Then, integrate iteratively (one variable at a time).

This process transitions smoothly to iterated integrals, which we'll delve into next.

Iterated Integrals

In cases involving double integrals, an iterated integral is a method that breaks down the process into two (or more) simple integrals. This approach not only makes complex calculations manageable but also essential where integrals depend on multiple variables.
In the iterated integral, you resolve the integration with respect to one variable, holding the other constant, and then iterate the process for the remaining variables. Let's look at how this works with an example from the problem:

• Consider the expression \(\int_{0}^{\sqrt{\pi} / 2} \int_{0}^{x^2} x \sec^{2} y \, dy \, dx \). Here, the setup has you first working on \(y\) over the given bounds, and then handling \(x\).
• Notice how we integrated \(\sec^2 y\) to find \(\tan y\), and then substituted for the limits before approaching the outer integral with \(x\) involved.

Each level of integration reduces complexity, transforming multi-variable integration into multiple one-dimensional integrals—if you follow carefully, the journey through the layers becomes straightforward.

Integration Techniques

Integration techniques are your toolbox, equipped with various strategies to tackle integrals. These techniques can include substitution, integration by parts, or using trigonometric identities.
In our exercise, we observed an approach directly utilizing the known integral for \( \sec^2 y \), which simplifies to \( \tan y \).This technique of recognizing direct results or using integral tables when a function fits a recognizable form is often an expedient way to resolve parts of double integrals.
Here are some general strategies:

• Substitution: Helpful in changing variables to simpler expressions, making the integral easier.
• Integration by Parts: Useful when the product of functions appears, this technique applies a rule related to derivatives and integrals.
• Partial Fraction Decomposition: Breaks rational expressions into simpler fractions—useful, although not shown here.

Every problem will call for a unique blend of techniques, and effective integration often involves trying multiple approaches.

Numerical Integration

Some integrals, like the final part of the exercise, resist solving via elementary functions. This is where numerical integration shines.
Numerical integration refers to techniques that approximate the value of an integral when an exact answer isn't feasible or practical. These methods are indispensible in calculus and applied fields due to their flexibility in handling complex or unsolvable integrands.
Methods often used include:

• Trapezoidal Rule: Uses trapezoids to approximate the area under a curve, splitting the interval into small sections.
• Simpson's Rule: Approximates the integrand by parabolic sections, giving good results for smooth functions.
• Midpoint Rule: Uses rectangles at midpoint heights, often more accurate than the trapezoidal rule in some contexts.

In our problem, using numerical techniques allows us to estimate the complex function \( \int_{0}^{\sqrt{\pi} / 2} x \tan(x^2) \, dx \), and is crucial for handling unwieldy integrals without closed-form solutions.