Problem 33 - Thomas Calculus

Question

In Exercises \(31-40,\) sketch the region of integration, reverse the order of integration, and evaluate the integral. $$ \int_{0}^{1} \int_{y}^{1} x^{2} e^{x y} d x d y $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2}e - 1 \).

Step 1: Sketch the Region of Integration

First, we identify the region of integration based on the limits of the given integral. The inner integral \( \int_{y}^{1} \cdots dx \) indicates that \( x \) ranges from \( y \) to \( 1 \). The outer integral \( \int_{0}^{1} \cdots dy \) shows that \( y \) ranges from \( 0 \) to \( 1 \). The region of integration is bounded by \( x = y \), \( x = 1 \), \( y = 0 \), and \( y = 1 \). This forms a triangular region in the coordinate plane.

Step 2: Identify the New Bounds for Reversed Order

To reverse the order of integration, we switch the roles of \( x \) and \( y \). Observing from the sketch, for a fixed \( x \), \( y \) ranges from \( 0 \) to \( x \), and \( x \) ranges from \( 0 \) to \( 1 \). Thus, the new limits of integration are: for \( x \) from \( 0 \) to \( 1 \), and for each \( x \), \( y \) varies from \( 0 \) to \( x \).

Step 3: Rewrite the Integral with the New Order

Rewrite the original integral with the reversed order of integration. The new integral becomes: \[ \int_{0}^{1} \int_{0}^{x} x^2 e^{xy} \, dy \, dx \]

Step 4: Evaluate the Inner Integral

Evaluate the inner integral with respect to \( y \): \[\int_{0}^{x} x^2 e^{xy} \, dy = x^2 \int_{0}^{x} e^{xy} \, dy \]The antiderivative of \( e^{xy} \) with respect to \( y \) is \( \frac{1}{x} e^{xy} \). Thus, the integral becomes: \[x^2 \left[ \frac{1}{x} e^{xy} \right]_{0}^{x} = x^2 \left( \frac{1}{x} e^{x \cdot x} - \frac{1}{x} e^{x \cdot 0} \right)\] Simplifying gives: \[x^2 \left( \frac{1}{x} (e^{x^2} - 1) \right) = x (e^{x^2} - 1)\]This simplifies further to: \[x e^{x^2} - x\]

Step 5: Evaluate the Outer Integral

Now evaluate the outer integral: \[ \int_{0}^{1} (x e^{x^2} - x) \, dx \]First, separate into two integrals: \[ \int_{0}^{1} x e^{x^2} \, dx - \int_{0}^{1} x \, dx \]For \( \int x e^{x^2} \, dx \), use substitution \( u = x^2 \), \( du = 2x \, dx \):\[\int x e^{x^2} \, dx = \frac{1}{2} \int e^{u} \, du = \frac{1}{2} e^{u}\]Back-substitute \( u = x^2 \) gives \( \frac{1}{2} e^{x^2} \). Evaluating from \( 0 \) to \( 1 \) gives:\[\frac{1}{2} (e^{1} - e^{0}) = \frac{1}{2} (e - 1)\]The \( \int_{0}^{1} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{1} = \frac{1}{2} \). Thus, combining everything: \[\frac{1}{2}(e - 1) - \frac{1}{2}\ = \frac{1}{2} e - 1\]

Step 6: Conclusion

The value of the given integral is therefore: \[ \frac{1}{2} e - 1 \].

Key Concepts

Reversing Order of Integration

Reversing the order of integration can be very helpful when dealing with double integrals. This technique involves interchanging the roles of the integration variables to potentially make computation easier. Here, it's like flipping the integration limits upside down or sideways, depending on how you look at it.
To do this, we must first sketch the region of integration described by the integral. This helps us visualize the range of each variable.
For the given problem, we initially have limits set by the outer integral: from 0 to 1 for \( y \), and the inner integral: from \( y \) to 1 for \( x \). This outlines a triangular region.
When reversing the order, notice that for each \( x \) within 0 and 1, \( y \) varies from 0 to \( x \). This makes our reversed order integral cleaner and potentially easier to evaluate.

Region of Integration

Understanding the region of integration is crucial for successfully evaluating double integrals.
The region boundaries tell us where our variables can "live" in the 2D plane.
For this problem, we sketch the region within the xy-plane bounded by the lines \( x = y \), \( x = 1 \), and \( y = 0 \), which forms a triangle.
These boundaries confine our integration to a specific area, ensuring that we calculate the integral only over the prescribed region. Visualizing this triangular area helps set up the integral correctly, whether performing the integration in the original or reversed order.
Properly identifying this region prevents errors and allows for a smoother evaluation process.

Integration Techniques

Double integration requires a systematic approach, leveraging various techniques to simplify computation.
When evaluating a given double integral, it is critical to tackle the inner integral first. This is often where things simplify considerably.
In this exercise, integrating \( x^2 e^{xy} \) with respect to \( y \) first optimizes the process. By finding the antiderivative, we reduce the complexity for the subsequent integral step.
It's crucial to break complex expressions into manageable parts, evaluating step-by-step. Also, separating integrals into distinct parts often clears a path to a solution, making convergence to the final answer more apparent.

Substitution in Integrals

Substitution is a powerful tool for simplifying integrals, especially when dealing with complex expressions.
The idea is to replace a difficult part of the integral with a simpler expression that is easier to integrate.
In our final step, we used substitution in \( \ \int x e^{x^2} \, dx \), letting \( u = x^2 \), which simplifies to handling the exponential term easily.
Through this substitution, and identifying \( du = 2x \, dx \), we converted a tricky iteration into a form where we have a standard integral of \( e^u \). This streamlined our computations significantly, showcasing the effectiveness of strategic substitution in deriving the final result.