Question

State why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral. $$ \int_{0}^{4} \int_{y / 2}^{2} e^{x^{2}} d x d y $$

Short answer

Direct integration is difficult due to \( e^{x^2} \); reorder and solve as \( \int_{0}^{2} \int_{2x}^{4} e^{x^2} \; dy \; dx \).

Step-by-step solution

Examine the Inner Integral

In the given integral \( \int_{0}^{4} \int_{y / 2}^{2} e^{x^2} \; dx \; dy \), evaluate the inner integral \( \int_{y/2}^{2} e^{x^2} \; dx \). The function \( e^{x^2} \) does not have a standard antiderivative, making it impossible to directly integrate with respect to \( x \).

Identify the Region of Integration

The limits \( x = y/2 \) and \( x = 2 \) with \( y \) from \( 0 \) to \( 4 \) describe a region in the \( xy \)-plane. Note that for each \( y \), \( x \) ranges from \( y/2 \) to \( 2 \). This means the region is bounded above by \( x = 2 \) and below by the line \( x = y/2 \).

Reverse the Order of Integration

To reverse the order, solve \( x = y/2 \) (i.e., \( y = 2x \)) to find \( y \) limits in terms of \( x \). The line \( x = y/2 \) corresponds to \( y = 2x \), so \( y \) varies from \( 2x \) to \( 4 \), while \( x \) varies from \( 0 \) to \( 2 \). The new integral is \( \int_{0}^{2} \int_{2x}^{4} e^{x^2} \; dy \; dx \).

Evaluate the New Inner Integral

With the new setup, the inner integral becomes \( \int_{2x}^{4} e^{x^2} \; dy \). As \( e^{x^2} \) is constant with respect to \( y \), the integral result is \( e^{x^2} (4 - 2x) \).

Evaluate the Outer Integral

The outer integral is now \( \int_{0}^{2} e^{x^2} (4 - 2x) \; dx \). This cannot be solved with elementary functions, so numerical methods or approximation techniques would be used in practice. However, the process has demonstrated reversing the integration order to simplify the problem conceptually.

Key concepts

Order of Integration

When dealing with iterated integrals, the order of integration refers to the sequence in which you integrate with respect to the variables. In a double integral, you have two options on which variable, say \( x \) or \( y \), to integrate first. The order of integration is specified by the limits of the integral.

• In the original integral \( \int_{0}^{4} \int_{y/2}^{2} e^{x^2} \; dx \; dy \), \( x \) is integrated first from \( y/2 \) to \( 2 \).
• The variable \( y \) is integrated next, ranging from \( 0 \) to \( 4 \).

Sometimes the initial order makes integration difficult or impossible if the function, like \( e^{x^2} \), lacks a standard antiderivative with respect to \( x \). This is when changing the order becomes advantageous. By switching the order of integration to \( \int_{0}^{2} \int_{2x}^{4} e^{x^2} \; dy \; dx \), we can simplify the process. Here, \( y \) is integrated first, keeping \( e^{x^2} \) constant, which aids in progressing toward a solution.

Integration Techniques

Integration techniques are methods used to evaluate integrals, especially when faced with functions that are challenging to integrate directly. The original problem involves the function \( e^{x^2} \), which does not have an elementary antiderivative. Traditional algebraic integration techniques such as substitution and integration by parts do not apply straightforwardly here.

• The primary technique demonstrated in the solution is changing the limits and variables. This allows the simplification of the integral as it permits \( e^{x^2} \) to act as a constant over the inner integral.
• Evaluating \( \int_{2x}^{4} e^{x^2} \; dy \) simplifies dramatically because integration with respect to \( y \) only involves multiplying \( e^{x^2} \) by the length of the interval \( 4 - 2x \).

Understanding when to apply a specific technique, like swapping integration order, is crucial in solving complex integrals that resist conventional methods.

Numerical Methods

Numerical methods come into play when integrals cannot be evaluated analytically using standard techniques. For the modified outer integral \( \int_{0}^{2} e^{x^2} (4 - 2x) \; dx \), no closed-form solution exists because \( e^{x^2} \) is problematic to integrate with functions like \( 4 - 2x \).

• Techniques like Simpson's Rule, Trapezoidal Rule, or numerical integration packages in software (like MATLAB or Python libraries) can approximate the integrals with desired accuracy.
• These methods are valuable for engineers and scientists who must compute solutions quickly without closed-form expressions.

The reliance on numerical solutions is not a deficiency but a practical approach aligned with real-world problem-solving where exact functions are less accessible.

Conceptual Simplification

The process of integration can sometimes feel daunting due to complex functions and variable limits. Conceptual simplification aims to reduce mental load by organizing information and steps logically. Key strategies include visualizing the region of integration and understanding geometric interpretations.

• Visualizing the region in the \( xy \)-plane initially given by \( x = y/2 \) to \( x = 2 \) helps contextualize the problem, revealing the bounds as a triangular region.
• Changing variables from \( x \) to \( y \), relabeling limits and understanding how the bounds translate geometrically simplifies the process.

Explaining why we reverse the order (here to make \( e^{x^2} \) independent of \( dy \)) alleviates confusion and streamlines problem-solving. Conceptual simplification allows learners to grasp the broader strategy, making the math less about computation and more about insight.