Problem 28 - Calculus Early Transcendentals

Question

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} x \sec ^{2} x y d A ; R=\\{(x, y): 0 \leq x \leq \pi / 3,0 \leq y \leq 1\\}$$

Step-by-Step Solution

Answer

Question: Determine the best order of integration for the iterated integral $$\iint_{R} x \sec ^{2} x y d A$$, where R is the region defined by 0 ≤ x ≤ π/3 and 0 ≤ y ≤ 1. Evaluate the integral using the best order determined. Answer: The best order for integration is dydx, and the integral evaluates to $$-\ln\frac{1}{2}$$.

Step 1: 1. Write the integral in order dx dy

$$\int_{0}^{1} \int_{0}^{\pi / 3} x \sec ^{2} (x y) dx dy$$ Next, let's examine this integral to determine if it would be easier to evaluate with respect to x first, or change the order of integration.

Step 2: 2. Analyze the integral to determine if it would be easier with respect to x first

When integrating with respect to x, there will be a trigonometric integral involving sec²(xy) and a factor of x. This may be difficult to evaluate directly. This suggests that it might be more convenient to integrate with respect to y first. Now let's change the order of integration:

Step 3: 3. Write the integral in the order dy dx

$$\int_{0}^{\pi / 3} \int_{0}^{1} x \sec ^{2} (x y) dy dx$$ With the new order of integration, we can now proceed to evaluate the integral.

Step 4: 4. Evaluate the inner integral (dy)

We first integrate with respect to y: $$\int_{0}^{1} x \sec ^{2} (x y) dy = \frac{x}{x} [\tan(xy)]_{0}^{1} = [\tan(x)] - [\tan(0)]$$ This simplifies to simply \(\tan(x)\). Now the integral becomes: $$\int_{0}^{\pi / 3} \tan(x) dx$$

Step 5: 5. Evaluate the outer integral (dx)

Finally, we integrate with respect to x: $$\int_{0}^{\pi / 3} \tan(x) dx = [-\ln|\cos(x)|]_{0}^{\pi / 3} = -\ln|\cos(\pi / 3)| + \ln|\cos(0)|$$ which simplifies to: $$-\ln\frac{1}{2} + \ln(1) = -\ln\frac{1}{2}$$ Thus, the integral is:

Step 6: Final answer

$$\iint_{R} x \sec ^{2} x y d A = -\ln\frac{1}{2}$$

Key Concepts

Trigonometric Integration

Trigonometric integration involves integrals of functions that use trigonometric identities and expressions. In our exercise, we have the term \( \sec^2(xy) \) which is a trigonometric function that often appears in integrals involving derivatives of tangent functions. When handling these functions, we need to leverage trigonometric identities to simplify and solve the integral efficiently.
Important to note is the derivative relationships:

• \( \frac{d}{dx} [\tan(x)] = \sec^2(x) \)
• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) can help in certain situations.

By understanding and using these identities, we made the integration process less cumbersome. We initially suspected challenges in evaluating the integral directly due to its trigonometric nature. However, switching the order of integration and recognizing the derivative-related identity allowed us to proceed smoothly.
Bringing these tools together, solving such integrals becomes significantly easier by restructuring and utilizing appropriate trigonometric properties.

Order of Integration

The order of integration in a double integral is crucial for simplifying calculations. In the given exercise, we initially explore the \( dx \ dy \) order, which involves directly integrating terms like \( x \sec^2(xy) \). This presents complications due to the intricate trigonometric functions when integrating with respect to \( x \) first.
Changing the order to \( dy \ dx \) simplifies the integral dramatically. This change allows us to tackle the trigonometric function first, leveraging the simplicity of the expression \( \sec^2(xy) \) when differentiated concerning \( y \).
The new order smoothly leads us to evaluate the integral as:

• Inner integral becomes \( \tan(x) \).
• Outer integral is relatively simple: finding the antiderivative of \( \tan(x) \).

By wisely choosing the order of integration, we eliminated complex intermediate steps, which showcases the importance of order in solving double integrals efficiently.

Double Integrals

Double integrals allow for evaluating functions over two-dimensional areas or regions. In this problem, we compute the double integral over the region \( R = \{(x, y): 0 \leq x \leq \frac{\pi}{3}, 0 \leq y \leq 1\} \). This region forms part of the rectangular plane.
We use iterated integrals to simplify our computations. An iterated integral breaks the process into two consecutive integrations: first over one variable, then the other. This approach requires careful consideration, such as checking bounds and ensuring the integration order optimizes computation difficulty.
Double integrals can be approached in either order (\(\int \int dx \ dy\) or \(\int \int dy \ dx\)), contingent upon which sequence simplifies the integration more effectively.
Ultimately, the double integral encapsulates finding the area under the curve within a specific region, balancing both the spatial restrictions and function's complexity. In our specific equation, by arranging the nested integrals correctly, we could effectively reduce and compute the complex integral into manageable parts.