Question

Many improper double integrals may be handled using the techniques for improper integrals in one variable. For example, under suitable conditions on \(f\) $$\int_{a}^{\infty} \int_{g(x)}^{h(x)} f(x, y) d y d x=\lim _{b \rightarrow \infty} \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) d y d x$$ Use or extend the one-variable methods for improper integrals to evaluate the following integrals. $$\int_{1}^{\infty} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x$$

Short answer

Question: Evaluate the improper integral $\int_{1}^{\infty} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x$. Answer: $-\frac{1}{4}$.

Step-by-step solution

Rewrite the integral as a limit

$$\int_{1}^{\infty} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x = \lim_{b \to \infty} \int_{1}^{b} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x$$ Now we proceed to evaluate the iterated integral.

Inner integral

First, we evaluate the inner integral: $$\int_{0}^{1 / x^{2}} \frac{2 y}{x} dy$$ As x is treated as a constant for this integral, the integral becomes $$\frac{2}{x} \int_{0}^{1 / x^{2}} y dy$$ Integrating with respect to y: $$\frac{2}{x} \left[\frac{1}{2}y^{2}\right]_{0}^{1 / x^{2}} = \frac{2}{x} \cdot \frac{1}{2x^{4}} = \frac{1}{x^{5}}$$

Evaluate the limit

Now our integral becomes, $$\lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{5}} dx$$

Outer integral

Integrating with respect to x, we get: $$\lim_{b \to \infty} \left[-\frac{1}{4x^4}\right]_{1}^{b}$$

Compute the limit

Now we compute the limit as b goes to infinity: $$\lim_{b \to \infty} \left(-\frac{1}{4b^4} - (-\frac{1}{4})\right)$$ As b goes to infinity, the first term goes to 0, leaving us with the final answer: $$-\frac{1}{4}$$ So, the value of the given improper integral is: $$\int_{1}^{\infty} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x = -\frac{1}{4}$$

Key concepts

Double Integrals

Double integrals allow us to calculate the volume under a surface defined by a function of two variables over a certain region. They extend the concept of single-variable integrals to two dimensions. When you're working with double integrals, think of slicing the region into infinitesimally small rectangles. The area of each small rectangle is multiplied by the height of the surface above it, given by the function value. This gives you a small volume element, and when all these small volumes are added up, you get the integral over the region.

For double integrals, you often see the notation \( \int \int_R f(x, y) \, dx \, dy \), where \(R\) is the region over which you're integrating. In many cases, such problems need to be solved using iterated integrals, which involves integrating with respect to one variable first and then the other.

Limits

In calculus, limits help us understand the behavior of functions as they approach certain points or even infinity. Limits are crucial in evaluating improper integrals, especially when the region of integration extends to infinity, as it does in this exercise. Here, we needed the concept of limits to handle the upper bound approaching infinity.

• The notation \( \lim_{b \to \infty} \) indicates we're taking the limit as \(b\) goes to infinity.
• This ensures the integral is evaluated not over an infinite distance directly but as a well-defined limit process.

By using limits, we can carefully analyze the behavior of a function at boundaries that extend towards infinity, facilitating the computation of otherwise impossible definite integrals.

Integration Techniques

Integration techniques simplify complex integrals into more manageable parts. In dealing with double integrals, you often perform integration iteratively. The solution involves several integration strategies:

• Changing the Order of Integration: Sometimes, switching the order of integration might simplify the process.
• Substitution: Useful when the integral contains composite functions.
• Integration by Parts: When products of functions are involved. Though not used in this specific problem, it plays a vital role in other contexts.
• Breaking Down into Iterated Integrals: As shown in the example, first solve the inner integral, then proceed to the outer integral.

Each technique has its purpose and utility, helping solve integrals that might otherwise seem impenetrable.

Iterated Integral

Iterated integrals break a double integral into two successive integrals. When working with iterated integrals, you'll solve one integral at a time, starting typically with the innermost one. The solution to the exercise involved this process:

1. **Inner Integral:** For a fixed \(x\), treat \(y\) as the only variable and find the integral \( \int_0^{1/x^2} \frac{2y}{x} \, dy \). 2. **Outer Integral:** Use the result of the inner integral and solve \( \int_1^{b} \frac{1}{x^5} \, dx \). 3. **Taking the Limit:** Finally, compute the limit of the outer integral as \(b\) approaches infinity. This step transforms the iterated integral from bounded to unbounded, addressing the behavior at infinity.The technique of iterated integration is invaluable, allowing sequential work with multiple variables, making the resolution of complex multivariable integrals possible.