Problem 292 - Calculus Volume 1

Question

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} x \sqrt{1-x^{2}} d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify a Suitable Substitution

Recognize that the integral involves a square root of the form \( \sqrt{1-x^2} \), which suggests using the trigonometric substitution \( x = \sin(\theta) \). This substitution simplifies \( \sqrt{1-x^2} \) into \( \sqrt{1 - \sin^2(\theta)} = \cos(\theta) \).

Step 2: Determine Differential and Limits of Integration

If \( x = \sin(\theta) \), then the differential \( dx \) is \( \cos(\theta) \, d\theta \). The original limits \( x = 0 \) and \( x = 1 \) are transformed to \( \theta = 0 \) (since \( \sin(0) = 0 \)) and \( \theta = \frac{\pi}{2} \) (since \( \sin(\frac{\pi}{2}) = 1 \)).

Step 3: Substitute into the Integral

Substitute \( x = \sin(\theta) \) and \( dx = \cos(\theta) \, d\theta \) into the integral, which now becomes \( \int_{0}^{\pi/2} \sin(\theta) \cos^2(\theta) \, d\theta \).

Step 4: Simplify the Integral with Trigonometric Identities

Use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) to rewrite the integral as \( \int_{0}^{\pi/2} \sin(\theta) \left( \frac{1 + \cos(2\theta)}{2} \right) d\theta \).

Step 5: Separate and Evaluate the Integrals

Separate the expression into two integrals: \( \frac{1}{2} \int_{0}^{\pi/2} \sin(\theta) \, d\theta + \frac{1}{2} \int_{0}^{\pi/2} \sin(\theta)\cos(2\theta) \, d\theta \). Evaluate the first integral, \( \int \sin(\theta) d\theta = -\cos(\theta) \), which gives \( \frac{1}{2} [ -\cos(\theta) ]_{0}^{\pi/2} = \frac{1}{2} (1-0) = \frac{1}{2} \).

Step 6: Evaluate the More Complex Integral

For the second integral, use the substitution \( u = 2\theta \), leading to \( du = 2 d\theta \), so the integral \( \int \sin(\theta) \cos(2\theta) d\theta \) requires integration by parts, or recognizing it as \( \int \frac{1}{2} \sin(2\theta) d\theta \), evaluating this from 0 to \( \pi/2 \) results in 0.

Step 7: Sum the Evaluated Integrals

Add the results from both integrals: \( \frac{1}{2} + 0 = \frac{1}{2} \).

Key Concepts

Definite Integral

A definite integral is all about calculating the exact area under a curve within a specified interval on the x-axis. This is different from an indefinite integral, which finds the general form of an antiderivative without bounds.
In our problem, the definite integral is \[ \int_{0}^{1} x \sqrt{1-x^{2}} d x \]The limits of integration, 0 to 1, are essential. They tell us where we start and stop measuring the area under the curve described by the function. When solving definite integrals, these boundaries provide a concrete result rather than a general form.
Definite integrals are vital tools in calculus, enabling us to find not only areas but also solutions related to accumulated quantities, like distance or volume.

Calculus Problem

This particular calculus problem involves evaluating a definite integral using substitution. Calculus problems often challenge students to manipulate and transform the given mathematical expressions.
Here, the core challenge is handling the square root in the integral \[ \sqrt{1-x^2} \]To solve such problems effectively, you need to not only understand integral calculus but also recognize patterns or forms that suggest specific techniques, like trigonometric substitution in this case.

• Breaking down complex expressions into manageable components.
• Choosing suitable methods for simplification.

These steps are a significant part of developing proficiency with calculus problems.

Integration by Substitution

Integration by substitution is a powerful technique that simplifies the process of finding an integral. Essentially, it involves changing variables to transform a complicated integral into a simpler or known form.
In our exercise, integration by substitution is kickstarted by noticing the expression \[ x = \sin(\theta) \]This clever substitution transforms the original integral into one involving sin and cos, two of the most straightforward trigonometric functions.
Then you replace every instance of the original variable and adjust the limits of integration to match the new variable.

• The differential changes too; here, from \[ dx = \cos(\theta) \, d\theta \]
• The boundaries alter from x-values to \[ \theta \] values, like from 0 to \[ \pi/2 \]

This method streamlines the integral evaluation process significantly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable involved. They're key to simplifying trigonometric integrals.
In this problem, we employ the identity \[ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \]This identity allows us to replace squared cosine functions with more simple terms, making them easier to integrate.

• Helps in transforming and breaking down complex expressions.
• Crucial in separating the integral into parts which can be individually evaluated.

Substitutions and identities work in tandem here. They simplify our integral so that it becomes manageable and eventually leads us to the correct solution step by step.