Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each of the iterated integrals. \(\int_{0}^{\pi} \int_{0}^{3} y \cos ^{2} x d y d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{9\pi}{4}\).

Step 1: Understand the Integral

The given integral is a double integral, which involves integrating a function over a rectangular region. The expression is \(\int_{0}^{\pi} \int_{0}^{3} y \cos ^{2} x \, d y \, d x\). This means we first integrate with respect to \(y\) and then integrate the resulting expression with respect to \(x\).

Step 2: Integrate with Respect to y

The inner integral is \(\int_{0}^{3} y \, d y\). Treat \(\cos^2 x\) as a constant with respect to \(y\). The integral of \(y\) with respect to \(y\) is \(\frac{y^2}{2}\). Thus, \(\int_{0}^{3} y \, d y = \left. \frac{y^2}{2} \right|_{0}^{3} = \frac{3^2}{2} - \frac{0^2}{2} = \frac{9}{2}\).

Step 3: Replace Result in the Outer Integral

We substitute the result from Step 2 into the outer integral. We now have the expression \(\int_{0}^{\pi} \frac{9}{2} \cos^2 x \, d x\).

Step 4: Simplify Cosine Square

Use the identity \(\cos^2 x = \frac{1 + \cos(2x)}{2}\) to simplify the integral. Therefore, the integral becomes \(\int_{0}^{\pi} \frac{9}{2} \times \frac{1 + \cos(2x)}{2} \, d x = \frac{9}{4} \int_{0}^{\pi} (1 + \cos(2x)) \, d x\).

Step 5: Integrate with Respect to x

Separate the integral: \(\frac{9}{4} \left( \int_{0}^{\pi} 1 \, d x + \int_{0}^{\pi} \cos(2x) \, d x \right)\). The first integral is \(\int_{0}^{\pi} 1 \, d x = [x]_{0}^{\pi} = \pi\). The second integral is \(\int_{0}^{\pi} \cos(2x) \, d x = \left. \frac{\sin(2x)}{2} \right|_{0}^{\pi} = 0\), as \(\sin(2\pi) = 0\) and \(\sin(0) = 0\).

Step 6: Calculate Final Result

Combine the results from Step 5 to get the final answer: \(\frac{9}{4} (\pi + 0) = \frac{9\pi}{4}\).

Key Concepts

Iterated Integral

Iterated integrals involve calculating integrals in a nested manner, one inside of another. In the given problem, the integral \( \int_{0}^{\pi} \int_{0}^{3} y \cos ^{2} x \ dy \ dx \) is an example of such an integral.

To solve it, you first handle the integral that is on the innermost part, which means you integrate with respect to one variable while keeping the other constant. This step-by-step process applies the so-called Fubini’s Theorem when working over rectangular regions.

When approaching iterated integrals, it's crucial to maintain the order of integration as indicated by the limits of each integral. This structured method allows us to break complex problems into simpler, manageable pieces.

Inner Integral

The concept of the inner integral is foundational in solving a double integral. In the provided solution, the inner integral is \( \int_{0}^{3} y \ dy \), where \( \cos^2 x \) is considered a constant and only \( y \) is treated as a variable.

Calculating the inner integral requires focusing on the given variable and integrating it over its specified limits. The integral of \( y \) with respect to \( y \) results in \( \frac{y^2}{2} \). Subsequently, by applying the limits from 0 to 3, we find \( \frac{9}{2} \).

This step isolates the effect of \( y \) and prepares the expression to be integrated with respect to \( x \) during the outer integration step, transforming a double integral into an iterated single integral.

Integration by Substitution

Integration by substitution is a powerful technique often used to simplify integrals. However, in this particular problem, the method isn’t directly needed for solving the integrals. Despite this, understanding this technique is vital in the toolbox of calculus strategies.

Generally, integration by substitution involves changing variables to simplify integration limits or integrands. When using this method, it's common to replace a complex variable or expression with a simpler one, perform the integral, and then back-substitute the original variable.

Although not applied here explicitly, this technique is an essential part of calculus that comes into play in different, more complex situations, particularly when direct integration is not immediately obvious.

Trigonometric Identities

Trigonometric identities often simplify the process of integration, especially when trigonometric functions like sine and cosine are involved.

In the solution, we used the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) to simplify the expression before integrating. This change turned the integral into something more manageable by reducing the complexity of the cosine-squared term.

When approaching integrals involving trigonometric functions, these identities can transform challenging integrals into ones that are straightforward to integrate, saving time and avoiding unnecessary complication during integration. Understanding and applying these identities can significantly enhance problem-solving efficiency.