Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-62, 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 the substitution variable

Here, let's choose the substitution variable. We notice that the integral contains terms \(\cos^2 x\) and \(\sin x\). A convenient choice would be to let \ u = \cos x\, thus making \ du = -\sin x \,dx\.

Step 2: Convert the limits of integration

Since we have substituted \ u = \cos x\, we need to convert the limits of integration: For the lower limit when \(x=0\), \(u = \cos 0 = 1\). For the upper limit when \(x=\frac{\pi}{2}\), \(u = \cos \frac{\pi}{2} = 0\).

Step 3: Perform the substitution

Now replace \(\cos x\) and \(\sin x \, dx\) in the integral with expressions in terms of \(u\): \[ \int_{0}^{\pi / 2} \cos^2 x \sin x \, dx = \int_{1}^{0} u^2 (-du) = \int_{0}^{1} u^2 \, du \]

Step 4: Evaluate the new integral

Now evaluate the integral \int_{0}^{1} u^2 \, du\:\[ = \left[\frac{u^3}{3}\right]_{0}^{1} = \left(\frac{1^3}{3}\right) - \left(\frac{0^3}{3}\right) = \frac{1}{3}. \]

Step 5: Conclusion

Thus, the definite integral \int_{0}^{\pi / 2} \cos^2 x \sin x \, dx\ evaluates to \(\frac{1}{3}\).

Key Concepts

Definite Integral

Definite integrals are a fundamental concept in calculus. They allow us to determine the accumulation of quantities, such as area, volume, and other physical properties, over an interval. In mathematical terms, a definite integral is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the integration limits, and \( f(x) \) is the function being integrated.
To comprehend this better, think of the definite integral as calculating the net area under the curve \( f(x) \) between the points \( x = a \) and \( x = b \). One notable aspect of definite integrals is that they yield a numerical value, a contrast to indefinite integrals, which result in a function plus a constant.

• The definite integral considers both positive and negative areas in the calculation.
• It requires evaluating the difference \( f(b) - f(a) \) to find the result.
• It's commonly shown using the Fundamental Theorem of Calculus.

Trigonometric Substitution

Trigonometric substitution is a clever technique used to simplify the integration of functions that involve trigonometric forms. By trading a trigonometric expression for a simpler form using substitution, we can make the integration process more manageable.
In our original exercise, we utilized this technique by substituting \( u = \cos x \). This substitution was ideal because it transforms \( \cos^2 x \) into a polynomial form, \( u^2 \), which is a simpler expression to integrate.

When to Use Trigonometric Substitution:

• Look for integrals involving \( \sin x \) and \( \cos x \), especially in powers.
• Consider substitution when the integral contains products or functions that resemble derivatives of trigonometric identities.
• This technique can also be useful in integrals involving \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), and \( \sqrt{x^2 - a^2} \).

By rewriting the integral in terms of \( u \), we can effectively simplify the integration process.

Integration Limits

The integration limits are the boundaries between which we calculate the definite integral. These limits are essential because they define the interval in which the integral will be evaluated.
When we perform a substitution like in our exercise, it's crucial to change the limits according to the new variable.
In the given problem:

• The original limits are \( x = 0 \) to \( x = \frac{\pi}{2} \).
• Substitution changes the variable to \( u = \cos x \), so the integration limits adjust accordingly.
• When \( x = 0 \), \( u = \cos 0 = 1 \).
• When \( x = \frac{\pi}{2} \), \( u = \cos \frac{\pi}{2} = 0 \).

This transformation of limits is crucial because it ensures that the integration accurately represents the area or value needed over the original interval.

Integral Evaluation

Integral evaluation involves computing the actual value that the definite integral represents. After setting up the integral correctly with the appropriate substitution or transformations, the next step is to solve it.
In our problem, once we used the trigonometric substitution, the integral turned into a simpler form \( \int_{0}^{1} u^2 \, du \), which is straightforward to evaluate.

Steps for Integral Evaluation:

• Integrate the simplified function: \[ \int u^2 \, du = \frac{u^3}{3} + C \] (indefinite form).
• Apply the limits to the integrated function: \[ \left[\frac{u^3}{3}\right]_{0}^{1} \]
• Calculate the difference: \( \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \).

This process confirms that the initial definite integral \( \int_{0}^{\pi / 2} \cos^2 x \sin x \, dx \) evaluates to \( \frac{1}{3} \), effectively completing the integration process.