Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \cos ^{2} x \sin x d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{3} \).

Step 1: Identify Substitution Components

To make the integration easier, identify the substitution. Notice that the integrand involves \( \cos^2 x \sin x \). We can choose \( u = \cos x \), which implies \( du = -\sin x \, dx \). This substitution will make the integral easier to evaluate.

Step 2: Adjust the Integral with Substitution

Substitute \( u = \cos x \) and adjust the differential and the limits of integration. If \( x = 0 \), then \( u = \cos 0 = 1 \). If \( x = \frac{\pi}{2} \), then \( u = \cos \frac{\pi}{2} = 0 \). Therefore, the integral becomes:\[ -\int_{1}^{0} u^2 \, du \]We change the limits appropriately when substituting the variable.

Step 3: Simplify the Integral

Since the integral was obtained with reversed boundaries, adjust it by switching the limits and changing the sign. Thus, we have:\[ \int_{0}^{1} u^2 \, du \]

Step 4: Calculate the Antiderivative

Find the antiderivative of \( u^2 \). The antiderivative is \( \frac{u^3}{3} \). So, the integral becomes:\[ \left[ \frac{u^3}{3} \right]_{0}^{1} \]

Step 5: Evaluate Definite Integral

Substitute the limits into the antiderivative to evaluate:\[ \left. \frac{u^3}{3} \right|_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]

Step 6: Write the Final Result

The evaluated definite integral from \( 0 \) to \( \frac{\pi}{2} \) for \( \cos^2 x \sin x \, dx \) is \( \frac{1}{3} \).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They represent the accumulation of quantities and allow us to calculate the total change over an interval. In essence, a definite integral computes the area under a curve defined by a function over a specific interval. In our exercise, we are interested in the definite integral of the function \( \cos^2 x \sin x \) over the interval \([0, \pi/2]\).

Here are some key things to remember about definite integrals:

• The limits of integration, denoted by the lower and upper bounds (here, \(0\) and \(\pi/2\)), signify the interval over which the function is integrated.
• The result of a definite integral is a number that represents the net area under the curve between the given bounds.
• Definite integrals can be evaluated using various techniques, including substitution, which is particularly useful when directly integrating a complex function is challenging.

By the end of calculating a definite integral, one typically has a specific value, like \(\frac{1}{3}\) in our problem, representing the total accumulated quantity over an interval.​

Substitution Method

The substitution method is a powerful technique for simplifying integrals. Often viewed as an inverse of the chain rule, substitution helps transform a complicated integral into a simpler one, making it more manageable to evaluate. For our exercise, the substitution method plays a crucial role in converting the integral of \( \cos^2 x \sin x \) into a more workable form.

Here's how substitution works in practice:

• First, choose a substitution that simplifies the integrand. In this case, selecting \( u = \cos x \) helps because it transforms \( \sin x \) into the differential \( du = -\sin x \, dx \).
• After substitution, adjust the limits of integration to reflect changes in the variable. For \( x = 0 \), \( u = 1 \), and for \( x = \pi/2 \), \( u = 0 \).
• Apply the new limits to obtain the integral \(-\int_{1}^{0} u^2 \, du\) and then simplify it by reversing the limits and changing the sign, resulting in \(\int_{0}^{1} u^2 \, du\).

Substitution is especially handy when dealing with products of functions and can often turn a challenging calculation into an easy one with a clear path forward. By simplifying the integrand, it becomes straightforward to proceed to the next step: calculating the antiderivative.

Antiderivative Calculation

The antiderivative calculation is the process of finding a function whose derivative yields the integrand. In our scenario, after simplifying the integral to \( \int_{0}^{1} u^2 \, du \), finding the antiderivative becomes our next focus. The antiderivative of a function reveals the accumulation of the function's values with respect to the variable. Here, we aim to compute the antiderivative of \( u^2 \).

Steps to find an antiderivative:

• Identify the basic form of the function. For \( u^n \), the antiderivative is \( \frac{u^{n+1}}{n+1} \). For our example, the antiderivative of \( u^2 \) is \( \frac{u^3}{3} \).
• After determining the antiderivative, apply the limits of integration. Using the expression \( \left[\frac{u^3}{3}\right]_{0}^{1} \), plug in the upper and lower bounds.
• Evaluate the expression at these bounds: \( \frac{1^3}{3} \) minus \( \frac{0^3}{3} \). This calculation results in the final answer, \( \frac{1}{3} \).

Thus, the process of calculating an antiderivative leads directly to evaluating the definite integral, tying all steps together to reach the conclusion given the function and limits.